New ternary self-orthogonal codes and related LCD
codes from weakly regular plateaued functions ∗†

arXiv:2312.04261v1 [cs.IT] 7 Dec 2023

Dengcheng Xie, Shixin Zhu‡, Yang Li
Abstract
A linear code is said to be self-orthogonal if it is contained in its dual. Selforthogonal codes are of interest because of their important applications, such as for
constructing linear complementary dual (LCD) codes and quantum codes. In this paper, we construct several new families of ternary self-orthogonal codes by employing
weakly regular plateaued functions. Their parameters and weight distributions are
completely determined. Then we apply these self-orthogonal codes to construct several new families of ternary LCD codes. As a consequence, we obtain many (almost)
optimal ternary self-orthogonal codes and LCD codes.

Keywords: Linear code, Self-orthogonal code, LCD code, Weakly regular plateaued function
Mathematics Subject Classification 94B05 12E10

1

Introduction

Throughout this paper, let p be an odd prime and n be a positive integer. Let Fq be the
finite field with q elements and F∗q = Fq \ {0}, where q = pn . A q-ary linear code C, denoted
by [n, k, d], is a k-dimensional linear subspace of Fnq and d is called the minimum Hamming
weight of C. Let the polynomial 1 + A1 z + · · · + An z n denote the weight enumerator of
C, where Ai is the number of codewords with Hamming weight i in C. Then the sequence
(1, A1 , A2 , · · · , An ) is called the weight distribution of C. If the number of i such that Ai 6= 0
in the sequence (1, A1 , A2 , · · · , An ) equals t, then the linear code C is called a t-weight code.
The well-known Sphere-packing bound [25] on a q-ary [n, k, d] linear code yields that
⌊(d−1)/2⌋  
X
n
n
k
(q − 1)i ,
(1)
q ≥q
i
i=0

where ⌊·⌋ denotes the floor function. An [n, k, d] linear code C is said to be optimal if there
is no [n, k, d′ ] linear code such that d′ > d and said to be almost optimal if there is an
optimal [n, k, d + 1] linear code.
∗

The authors are with School of Mathematics, Hefei University of Technology, Hefei, 230601, China
(email: dengchengxiemath@163.com, zhushixinmath@hfut.edu.cn, yanglimath@163.com).
†
This research is supported by the National Natural Science Foundation of China (Grant Nos.
U21A20428 and 12171134).
‡
Corresponding author

1

1.1

Self-orthogonal codes and linear complementary dual codes
For any q-ary [n, k, d] linear code, its dual code is a linear code defined by

C ⊥ = c⊥ ∈ Fnq : hc⊥ , ci = 0 for all c ∈ C

and C ⊥ has parameters [n, n − k], where h·, ·i denotes the standard inner product. We
call C a self-orthogonal code if C ⊆ C ⊥ . Self-orthogonal codes have been widely studied
since coding theory was proposed [42]. There are several reasons why they attracted wide
interest. On the one hand, they are closely connected with other mathematical structures
such as combinatorial t-design theory [1], group theory [11] and modular forms [45]. On
the other hand, they have vital applications in constructing quantum codes [5, 9, 36, 37, 48].
Note also that there are too many self-orthogonal codes as the dimension k increases and
the inner product changes. Li et al. [33] and Shi et al. [44] determined the minimum
distance of an optimal binary Euclidean self-orthogonal [n, k] code for k ≤ 6. Bouyukliev
et al. [4] also classified ternary Euclidean self-orthogonal codes and quaternary Hermitian
self-orthogonal code for small parameters. For more details on self-orthogonal codes, we
refer to [3, 27, 28, 41, 52] and the references therein.
A linear code C is said to be linear complementary dual (LCD) if C ∩ C ⊥ = {0}, where
0 is a zero vector. With [38], an [n, k] linear code C generated by a matrix G is LCD if
and only if GGT is nonsingular. It is also clear that the dual code of an LCD code is still
LCD. At first, LCD codes were introduced by Massey [38] in order to solve a problem in
information theory. Sendrier [43] showed that LCD codes meet the asymptotic GilbertVarshamov bound. Carlet and Guilley [6] investigated an application of binary LCD codes
against Side-Channel Attacks (SCAs) and Fault Injection Attacks (FIAs). The study of
LCD codes has thus become a hot research topic and been further carried out recently
(see [7, 16, 24, 26, 29, 31, 32, 34, 46]). In particular, Carlet et al. showed that any q-ary
linear code is equivalent to some Euclidean LCD code in [8]. This motivates us to study
LCD codes over small finite fields. Therefore, it is always very interesting to construct
self-orthogonal codes and LCD codes in the study of coding theory.
1.2

Related works on the constructions of linear codes

It is well-known that there are many different methods to construct linear codes. On the
one hand, we note that one of them is the so-called generic method, which was proposed
by Ding et al. in [13]. Specifically, we consider the absolute trace function T rpn (x) =
2
n−1
that maps an element x ∈ Fq to another element of Fp . By fixing
x + xp + xp + · · · + xp
a set D = {d1 , d2 , · · · , dn } ⊆ F∗q , Ding et al. [13] proposed a general method to generate
′
linear codes CD
with the form of


′
CD
= { T rpn (xd1 ), T rpn (xd2 ), · · · , T rpn (xdn ) : x ∈ Fq }.
(2)

′
Furthermore, the set D is called the defining set of CD
. In [13], Ding et al. chose the
∗
2
defining set D = {x ∈ Fq : x = 0} and constructed several new families of 2-weight and
3-weight linear codes. Based on this generic method, many researchers have employed pary (weakly regular) bent functions and plateaued functions to provide defining sets and

2

constructed infinite families of linear codes with few weights or other desired properties
(see for example [12, 14, 15, 19, 20, 22, 23, 40, 49–51]).
Note that compared with using only one weakly regular bent or plateaued function,
it remains an open problem for a long time to construct linear codes with more flexible
parameters by mixing two or more (possibly) different functions. In 2021, Cheng et al.
fulfilled this gap in [10]. They presented a new construction of linear codes, which gives a
linear code
CD = {T rp2n (αx + βy)(x,y)∈D : α, β ∈ Fq },

(3)

where D = {(x, y) ∈ Fq × Fq \{(0, 0)}, f (x) + g(x) = c} for c ∈ F∗p and f, g : Fq → Fp are
weakly regular plateaued functions. As a result, they obtained several classes of few-weight
linear codes with good and flexible parameters by using the properties of cyclotomic fields
and exponential sums.
Very recently, by employing the well-known augmented construction [25], Heng et al. [21]
′
considered the augmented code of CD
defined in Equation (2), which has the form of


′
CD
= { T rpn (xd1 ), T rpn (xd2 ), · · · , T rpn (xdn ) + µ1 : x ∈ Fq , µ ∈ Fp },
(4)

where the defining set D = {d1 , d2, · · · , dn } ⊆ F∗q and 1 = (1, 1, · · · , 1) ∈ Fnp . This
′
construction can yield linear code with dimension increasing by 1 if 1 ∈
/ CD
and hence,
Heng et al. obtained several families of ternary self-orthogonal codes of larger dimensions
by choosing defining sets from bent functions. They also further constructed related ternary
LCD codes from these self-orthogonal codes.
1.3

Motivations and contributions

Note that weakly regular plateaued functions are generalizations of bent functions.
Inspired by the ideas and methods in [10,21], a natural and interesting problem rises: Can
new infinite families of self-orthogonal codes and LCD codes are constructed
via the augmented codes of CD , where the defining sets D are chosen by mixing
two different weakly regular plateaued functions?
Motivated by this problem, we study in this paper the ternary augmented code

(5)
CD = T r3s (αx + βy)(x,y)∈D + µ1 : (α, β) ∈ F3n × F3m , µ ∈ F3 ,
where s = n + m with n and m being two positive integers and the defining set D is chosen
from two different weakly regular plateaued functions. According to [10], we know that
1∈
/ CD , and hence, the augmented construction CD is meaningful. Specifically, for λ ∈ F∗3 ,
we will consider the following two new defining sets:
Df g (0) = {(x, y) ∈ F3n × F3m : f (x) + g(y) + λ = 0} and
Dg (0) = {(x, y) ∈ F3n × F3m : T r3n (x) + g(y) + λ = 0} ,

(6)
(7)

where f (x) and g(y) are respectively weakly regular kf -plateaued and kg -plateaued unbalanced functions, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m.
Moreover, our main contributions can be summarized as follows:
3

(1) Based on these two new defining sets, we obtain several new infinite families of ternary
self-orthogonal codes in Theorems 4.2, 4.4, 4.8 and 4.9, respectively. We completely
determine their weight distributions and list respectively the weight distributions in
Tables 1-5. We also determine the whole parameters of the dual codes of these selforthogonal codes in Theorems 5.1 and 5.2. As explicit examples, we list some (almost)
optimal ternary self-orthogonal codes in Examples 4.10 and 4.11.
(2) Based on the ternary self-orthogonal codes given above, we further construct several
new families of related ternary LCD codes. We completely determine the whole
parameters of these LCD codes as well as their dual codes in Theorems 6.3-6.5. It
is worth noting that they contain several new infinite families of ternary LCD codes,
which are at least almost optimal with respect to the Sphere-packing bound given in
Equation (1).
The paper is organized as follows. In Section 2, we review some useful basic knowledge
on cyclotomic fields, weakly regular plateaued functions and the Pless power moments. In
Section 3, we give some auxiliary results for later use. In Section 4, we construct several
new infinite families of ternary self-orthogonal codes. In Section 6, we investigate the dual
codes of these ternary self-orthogonal codes. In Section 6, we consider an application of
ternary self-orthogonal codes in ternary LCD codes and derive several new infinite families
of ternary LCD codes that contain (almost) optimal ternary LCD codes. In Section 7, we
conclude this paper.

2

Preliminaries

In this section, we recall some basic knowledge on cyclotomic fields, weakly regular
plateaued functions and Pless power moments.
2.1

Cyclotomic fields

Let p be an odd prime and ζp be a primitive p-th complex root of unity. Let SQ and
N SQ denote the set of all nonzero squares and nonsquares in Fp , respectively. Let η0 be
p−1
the quadratic characters of F∗p and p∗ = η0 (−1)p = (−1) 2 p. The following lemma are
deduced from [35].
Lemma 2.1 ( [35]). Let notations be the same as above. Then we have
P
(1)
κ∈F∗p η0 (κ) = 0;
P
√ ∗
κ
p;
(2)
κ∈F∗p η0 (κ)ζp =
(3)

P

κτ
κ∈F∗p ζp = 1 and

P

κ2 τ
= η0 (τ )
κ∈Fp ζp

√ ∗
p for any τ ∈ F∗p .

A cyclotomic field Q(ζp ) is obtained from the rational field Q by adjoining ζp . The field
extension Q(ζp )/Q is Galois of degree p − 1 and the Galois group of Q(ζp ) over Q is
Gal(Q(ζp )/Q) = {σκ : κ ∈ F∗p },
4

where σκ is an automorphism of Q(ζp ) defined by σκ (ζp ) = ζpκ . For any κ ∈ F∗p and τ ∈ Fp ,
√ n
√ n
we clearly have σκ (ζpτ ) = ζpκτ and it follows from Lemma 2.1 that σκ (√ p∗ ) = η0n (κ) p∗ .
Hence, the cyclotomic field Q(ζp ) has a unique quadratic subfield Q( p∗ ) and
√ 
 

Gal Q p∗ /Q = {1, σµ } for µ ∈ N SQ.
2.2

Weakly regular plateaued functions

Definition 2.2. Assume that f : Fq → Fp be a p-ary function and α ∈ Fq , then the Walsh
transform of a p-ary function f is given by
X f (x)−T rn (αx)
p
e f (α) =
ζp
,
(8)
R
x∈Fpn

where ζp is a primitive p-th complex root of unity.

e f (0) = 0 and bent if |R
e f (α)| = p n2 for every
A p-ary function f is called balanced if R
e f (α) = √pn ζpf ∗ (α) and called weakly regular
α ∈ Fq . A bent function f is called regular if R
∗
e f (α) = ǫ√pn ζpf (α) for all α ∈ Fq , where
if there is a complex root of unity ǫ such that R
f ∗ (x) is a q-ary function from Fq to Fp and called the dual of f (x). Moreover, f ∗ (x) is also
a weakly regular bent function.
As generalizations of bent functions, plateaued functions were introduced over finite
fields with characteristic two in [53]. Specifically, a p-ary function f : Fq → Fp is called
e f (α)|2 ∈ {0, pn+kf } for every α ∈ Fq , where kf is an integer satisfying
kf -plateaued if |R
0 ≤ kf ≤ n. It is obvious that every bent function coincides with a 0-plateaued function.
The Walsh support of a kf -plateaued function f is given by
g f = {α ∈ Fq : |R
e f (α)|2 = pn+kf }.
SR

(9)

g f |=pn−kf , which checks the following proposition.
Clearly, |SR

Proposition 2.3. Let f be a kf -plateaued function from Fq to Fp , where kf is an integer
satisfying 0 ≤ kf ≤ n. Then there exist pn−kf (resp. pn − pn−kf ) different α ∈ Fq such that
e f (α)|2 = pn+kf (resp. 0).
|R

Definition 2.4. Let f be a kf -plateaued function from Fq to Fp , where kf is an integer
satisfying 0 ≤ kf ≤ n. Then f is said to be weakly regular kf -plateaued if there exists a
complex root of unity u such that
e f (α) ∈ {0, up
R

n+kf
2

ζpf (α) }
∗

(10)

gf .
for all α ∈ Fq , where f ∗ is a p-ary function over Fq with f ∗ (α) = 0 for all α ∈ Fq \ SR

Let εf = ±1 be a sign related to the Walsh transform of a p-ary function f . Mesnager
et al. [40] and Sinak et al. [47] presented the following result.
5

Proposition 2.5 ( [40, 47]). Let f : Fq → Fp be a weakly regular kf -plateaued function,
where kf is an integer with 0 ≤ kf ≤ n. Let WRP be the non-trivial class of weakly regular
unbalanced kf -plateaued functions f satisfying the following homogeneous conditions:
(1) f (0) = 0;
(2) f (ax) = ahf f (x) for every a ∈ F∗p and x ∈ Fq , where hf is an even positive integer
such that gcd(hf − 1, p − 1) = 1.
∗
e f (α) = εg √p∗ n+kf ζpf (α) for every α ∈ SR
g g , there exists an
Then if f ∈ WRP with R
g g , where
even positive integer lf such that f ∗ (bα) = blf f ∗ (α) for any b ∈ F∗p and α ∈ SR
gcd(lf − 1, p − 1) = 1.

2.3

Pless power moments

⊥
⊥
For any p-ary [n, k, d] linear code, denote by (1, A1 , A2 , · · · , An ) and (1, A⊥
1 , A2 , · · · , An )
the weight distribution of C and C ⊥ , respectively. The following well-knowledge results are
called the first four Pless power moments [25]:

P1 :
P2 :
P3 :

n
X

j=0
n
X

j=0
n
X
j=0

P4 :

Aj =q k ,

n
X
j=0

jAj =q k (qn − n − A⊥
1 ),



⊥
j 2 Aj =q k−2 (q − 1)n(qn − n + 1) − (2qn − q − 2n + 2)A⊥
1 + 2A2 ,

j 3 Aj =q k−3 [(q − 1)n(q 2 n2 − 2qn2 + 3qn − q + n2 − 3n + 2)
− (3q 2 n2 − 3q 2 n − 6qn2 + 12qn + q 2 − 6q + 3n2 − 9n + 6)A⊥
1
⊥
+ 6(qn − q − n + 2)A⊥
2 − 6A3 ].

It should be emphasized that these four Pless power moments play important roles in
determining the minimum weight of the dual code of a linear code.

3

Some auxiliary results

√
√ ∗
From this section on, we
√ let p = 3. Then p = −3 and η0 (−1) = −1. Note that
we still use the notations 3∗ and η0 (−1) in the sequel for the uniform representations.
In the following, we present some auxiliary results, which will be used for determining the
parameters and weight distributions of ternary linear codes in the next section.
6

√
∗
g f and
e f (α) = εf 3∗ n+kf ζ3f (α) for every α ∈ SR
Lemma 3.1. Let f, g ∈ WRP with R
√ m+kg g∗ (β)
e g (β) = εg 3∗
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and
R
ζ3
for every β ∈ SR
0 ≤ kg ≤ m. Let s = n + m, λ ∈ F3 and
g f × SR
g g : f ∗ (α) + g ∗ (β) + λ = 0}.
Pf ∗ g∗ (0) = #{(α, β) ∈ SR

Then the following statements hold.
(1) If s + kf + kg is even, then


s−k −kg −2
3s−kf −kg −1 + 2(−1)s+1 ε ε (−3) f 2
λ = 0,
f g
Pf ∗ g∗ (0) =
s−kf −kg −2
 s−kf −kg −1
2
3
− (−1)s+1 εf εg (−3)
λ ∈ F∗3 .

(2) If s + kf + kg is odd, then


3s−kf −kg −1
λ = 0,



s−kf −kg −1
2
Pf ∗ g∗ (0) = 3s−kf −kg −1 + (−1)s+1 εf εg (−3)
λ = 1,


s−k −k −1

3s−kf −kg −1 − (−1)s+1 ε ε (−3) f 2 g
λ = 2.
f g

Proof. By the definition of Pf ∗ g∗ (0), we have
1X X
3 t∈F

X

= 3s−kf −kg −1 +

1X

Pf ∗ g∗ (0) =

t(f ∗ (α)+g ∗ (β)+λ)

ζ3

gg
g f β∈SR
3 α∈SR

3

t∈F∗3



ζ3λt σt 

X

gf
α∈SR

f ∗ (α)

ζ3

X

gg
β∈SR

g ∗ (β)

ζ3




√ s−kf −kg −2 X s−kf −kg
(t)ζ3λt .
η0
= 3s−kf −kg −1 + (−1)s+1 εf εg 3∗
t∈F∗3

Then the desired results follow straightforward from Lemma 2.1.
√
∗
g f and
e f (α) = εf 3∗ n+kf ζ3f (α) for every α ∈ SR
Lemma 3.2. Let f, g ∈ WRP with R
√ m+kg g∗ (β)
e g (β) = εg 3∗
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and
R
ζ3
for every β ∈ SR
0 ≤ kg ≤ m. Let s = n + m, λ ∈ F∗3 ,
g f × SR
g g : f ∗ (α) + g ∗ (β) + λ = 1} and
Pf ∗ g∗ (1) = #{(α, β) ∈ SR
g f × SR
g g : f ∗ (α) + g ∗ (β) + λ = 2}.
Pf ∗ g∗ (2) = #{(α, β) ∈ SR

Then the following statements hold.

7

(1) If s + kf + kg is even, then
s−kf −kg
1 + 3η0 (λ)
and
(−1)s εf εg (−3) 2
6
s−kf −kg
1 − 3η0 (λ)
(−1)s εf εg (−3) 2 .
Pf ∗ g∗ (2) = 3s−kf −kg −1 +
6

Pf ∗ g∗ (1) = 3s−kf −kg −1 +

(2) If s + kf + kg is odd, then
s−kf −kg −1
1 + η0 (−λ)
2
and
(−1)s εf εg (−3)
2
s−kf −kg −1
1 − η0 (−λ)
2
(−1)s εf εg (−3)
Pf ∗ g∗ (2) = 3s−kf −kg −1 +
.
2

Pf ∗ g∗ (1) = 3s−kf −kg −1 −

Proof. Similar to Lemma 3.1, we still denote
n
o
∗
∗
g
g
Pf ∗ g∗ (0) = # (α, β) ∈ SRf × SRg : f (α) + g (β) + λ = 0 .
Consider the exponential sum
X X
P =

X

t2 (f ∗ (α)+g ∗ (β)+λ)

ζ3

t∈F3 α∈SR
gg
g f β∈SR

=

X

X

gg
g f β∈SR
α∈SR

= 3s−kf −kg +




X

t2 (f ∗ (α)+g ∗ (β)+λ)

ζ3

t∈F∗3

X

t∈F∗3

2



ζ3λt σt2 

= 3s−kf −kg + (−1)s εf εg 3∗
= 3s−kf −kg + (−1)s εf εg

+ 1

f ∗ (α)

ζ3

gf
α∈SR

√

√

X



s−kf −kg X

X

gg
β∈SR
2

ζ3λt



g ∗ (β) 

ζ3

t∈F∗

 3 √

s−kf −kg
∗
∗
η0 (λ) 3 − 1 .
3

It follows from the definitions of Pf ∗ g∗ (1) and Pf ∗ g∗ (2) that
√
√
P = 3Pf ∗ g∗ (0) + 3∗ Pf ∗ g∗ (1) − 3∗ Pf ∗ g∗ (2).

Combining the fact that Pf ∗ g∗ (0) + Pf ∗ g∗ (1) + Pf ∗ g∗ (2) = 3s−kf −kg , the desired results follow
immediately straightforward from Lemma 3.1.
√
g g , where
e g (β) = εg 3∗ m+kg ζ g∗ (β) for every β ∈ SR
Lemma 3.3. Let g ∈ WRP with R
3
∗
εg ∈ {−1, 1} and 0 ≤ kg ≤ m. Let λ ∈ F3 , µ ∈ F3 and
g g : g ∗(β) + λ −
Pg∗ (0) = #{α ∈ F∗3 , β ∈ SR

Then the following statements hold.

8

µ
= 0}.
α

(1) If m + kg is even, then

m−k −2
2 · 3m−kg −1 − 2 · (−1)m+1 ε (−3) 2g
µ = 0,
g
Pg∗ (0) =
m−kg −2
2 · 3m−kg −1 + (−1)m+1 ε (−3) 2
µ=
6 0.
g

(2) If m + kg is odd, then

m−k −1
2 · 3m−kg −1 + 2 · (−1)m+1 ε (−3) 2g η (λ) µ = 0,
g
0
Pg∗ (0) =
m−kg −1
2 · 3m−kg −1 + (−1)m+1 ε (−3) 2 η (2λ)
µ=
6 0.
g
0

Proof. By the definition of Pg∗ (0), we have
1 X X X t(g∗ ( αβ )+λ− αµ )
ζ3
Pg∗ (0) =
3 t∈F α∈F∗
g
3
3 β∈SRg


X
X
X
β
µ
∗
1
g ( )+λ− α

ζ3 α
σt 
= 2 · 3m−kg −1 +
3 t∈F∗
α∈F∗3 β∈SR
gg
3


X
X
X
1
g ∗ (β)+λ−µ
g ∗ (β)+λ+µ 
= 2 · 3m−kg −1 +
σt 
ζ3
+
ζ3
,
3 ∗
t∈F3

gg
β∈SR

gg
β∈SR

where the last equation holds since α ∈ F∗3 .
For µ = 0, we further have


X
X
∗
2
g (β)+λ 
ζ3
σt 
Pg∗ (0) = 2 · 3m−kg −1 +
3 t∈F∗
gg
β∈SR

3

= 2 · 3m−kg −1 + 2 · (−1)m+1 εg (−3)

m−kg −2
2

X

m−kg

η0

(t)ζ3λt

t∈F∗

3

m−kg −2
2 · 3m−kg −1 − 2 · (−1)m+1 ε (−3) 2
m + kg is even,
g
=
m−kg −1
2 · 3m−kg −1 + 2 · (−1)m+1 ε (−3) 2 η (λ) m + k is odd.
g
0
g

For µ 6= 0, it can also be checked that
Pg∗ (0) = 2 · 3m−kg −1 +

1X

3 t∈F∗
3



σt 

X

g ∗ (β)

ζ3

gg
β∈SR

= 2 · 3m−kg −1 + (−1)m+1 εg (−3)

m−kg −2
2

+

X

gg
β∈SR

X

t∈F∗

g ∗ (β)+2λ 

ζ3

m−kg

η0



(t) 1 + ζ32λt




3

m−kg −2
m−k
−1
m+1
2 · 3
g
+ (−1)
εg (−3) 2
m + kg is even,
=
m−kg −1
2 · 3m−kg −1 + (−1)m+1 ε (−3) 2 η (2λ) m + k is odd.
0
g
g

9

We have finished the proof.
√
g f and
e f (α) = εf 3∗ n+kf ζ f ∗ (α) for every α ∈ SR
Lemma 3.4. Let f, g ∈ WRP with R
3
√ m+kg g∗ (β)
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and
e g (β) = εg 3∗
ζ3
for every β ∈ SR
R
∗
0 ≤ kg ≤ m. Let s = n + m, λ ∈ F3 and
X
X
t(f (x)+g(y)+λ)
ζ3
.
S1 =
t∈F∗3 (x,y)∈F3n ×F3m

Then we have
S1 =
Proof. It is clear that


s+k +kg
 − ε ε (−3) f2

s + kf + kg is even,

f g


εf εg (−3)

s+kf +kg +1
2

X

S1 =

X

η0 (λ) s + kf + kg is odd.
t(f (x)+g(y)+λ)

ζ3

t∈F∗3 (x,y)∈F3n ×F3m

X

=

ζ3λt σt

=

f (x)
ζ3

x∈F3n

t∈F∗3

X

X

√

ζ3λt εf εg 3∗

X

g(x)
ζ3

y∈F3m

!

s+kf +kg s+kf +kg
η0
(t).

t∈F∗3

The desired result then follows from Lemma 2.1.
Lemma 3.5. Let s = n + m and µ ∈ F3 , assume that
X
X
u(T r s (αx+βy)+µ)
ζ3 3
.
S2 =
u∈F∗3 (x,y)∈F3n ×F3m

For any (α, β) ∈ F3n × F3m , we have

(α, β) 6= (0, 0),

0
S2 = 2 · 3s (α, β) = (0, 0) and µ = 0,


− 3s (α, β) = (0, 0) and µ 6= 0.
Proof. It can be calculated that
X
S2 =

X

u(T r3s (αx+βy)+µ)

ζ3

u∈F∗3 (x,y)∈F3n ×F3m

=

X

u∈F∗3

ζ3µu σu

X

T r n (αx)
ζ3 3

x∈F3n

X

y∈F3m

T r m (βy)
ζ3 3


(α, β) 6= (0, 0),

0
s
= 2 · 3 (α, β) = (0, 0) and µ = 0,


− 3s (α, β) = (0, 0) and µ 6= 0.

This completes the proof.

10

!

Lemma 3.6. Let notations be the same as Lemma 3.4. Let lf and lg be two even positive
g f and
integers satisfying f ∗ (aα) = alf f ∗ (α) and g ∗ (bβ) = blg g ∗ (β), where a, b ∈ F∗3 , α ∈ SR
g g . Let s = n + m, λ ∈ F∗ , µ ∈ F3 and
β ∈ SR
3
S3 =

XX

X

s

ζpt(f (x)+g(y)+λ)+u(T r3 (αx+βy)+µ) .

t∈F∗3 u∈F∗3 (x,y)∈F3n ×F3m

g f × SR
g g , then S3 = 0. If (α, β) ∈ SR
g f × SR
g g , then the following statements
If (α, β) ∈
/ SR
hold.
(1) If s + kf + kg is even, then

s+kf +kg

2

4ε
ε
(−3)

f g




s+kf +kg
S3 = − 2εf εg (−3) 2






ε ε (−3) s+kf2+kg
f g

f ∗ (α) + g ∗ (β) + λ = 0 and µ = 0,
f ∗ (α) + g ∗ (β) + λ = 0 and µ 6= 0,
or f ∗ (α) + g ∗(β) + λ 6= 0 and µ = 0,
f ∗ (α) + g ∗ (β) + λ 6= 0 and µ 6= 0.

(2) If s + kf + kg is odd, then


0



s+kf +kg +1



2

2ε
ε
(−3)
f
g



s+kf +kg +1
2
S3 = − 2εf εg (−3)


s+kf +kg +1


2

−
ε
ε
(−3)
f
g



s+kf +kg +1


2
εf εg (−3)

Proof. It is clear that
XX
S3 =

X

f ∗ (α) + g ∗ (β) + λ = 0,
f ∗ (α) + g ∗ (β) + λ = 1 and µ = 0,
f ∗ (α) + g ∗ (β) + λ = 2 and µ = 0,
f ∗ (α) + g ∗ (β) + λ = 1 and µ 6= 0,
f ∗ (α) + g ∗ (β) + λ = 2 and µ 6= 0.

t(f (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)

ζ3

t∈F∗3 u∈F∗3 (x,y)∈F3n ×F3m

=

X

u∈F∗3

=

X

u∈F∗3

X
µu

ζ3

t∈F∗3

f (x)−T r3n (− ut αx)
ζ3

x∈F3n

X

y∈F3m



e f (− u α)R
e f (− u α) .
ζ3λt σt R
t
t
t∈F∗

X
µu

ζ3

ζ3λt σt

X

g(x)−T r3m (− ut βx)
ζ3

!

3

g f × SR
g g and hence, S3 = 0;
g f × SR
g g , then (− uα , − uβ ) ∈
/ SR
For t, u ∈ F∗3 , if (α, β) ∈
/ SR
t
t
g f × SR
g g , then (− uα , − uβ ) ∈ SR
g f × SR
g g and it follows from the facts
and if (α, β) ∈ SR
t
t
u lf ∗
u lg ∗
∗
that 2 | lf and 2 | lg that (− t ) f (α) = f (α) and (− t ) g (β) = g ∗(β). It then implies
11

that
S3 =

X

ζ3−µu

t∈F∗3

u∈F∗3

=

X

X

ζ3−µu

u∈F∗

X

l
√ s+kf +kg s+kf +kg
t((− u ) f f ∗ (α)+(− ut )lg g ∗ (β))
ε f ε g 3∗
η0
(t)ζ3λt ζ3 t

√ s+kf +kg s+kf +kg
t(f ∗ (α)+g ∗ (β)+λ)
ε f ε g 3∗
(t)ζ3
η0

t∈F∗

3
 3
s+kf +kg

2

4ε
ε
(−3)

f g




s+kf +kg
= − 2εf εg (−3) 2






ε ε (−3) s+kf2+kg

f g

f ∗ (α) + g ∗(β) + λ = 0 and µ = 0,
f ∗ (α) + g ∗(β) + λ = 0 and µ 6= 0,
or f ∗ (α) + g ∗ (β) + λ 6= 0 and µ = 0,
f ∗ (α) + g ∗(β) + λ 6= 0 and µ 6= 0,

when s + kf + kg is even. This completes the desired result (1). Taking the same argument
as that of the proof of the result (1), the expected result (2) also holds. We have finished
the whole proof.
Lemma 3.7. Let notations be the same as Lemma 3.3. Let s = n + m, λ ∈ F∗3 , µ ∈ F3 and
S4 =

XX

X

t(T r3n (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)

ζ3

.

t∈F∗3 u∈F∗3 (x,y)∈F3n ×F3m

For any (α, β) ∈ F3n × F3m \{(0, 0)}, the following statements hold.
(1) If m + kg is even, then

S4 =


m+kg

n
2

2
·
3
ε
(−3)

g


n

− 3 εg (−3)





0

m+kg
2

g g and g ∗ ( β ) + λ − µ = 0,
α ∈ F∗3 , β ∈ SR
α
α
β
g g and g ∗ ( ) + λ − µ 6= 0,
α ∈ F∗3 , β ∈ SR
α
α
otherwise.

(2) If m + kg is odd, then

S4 =


m+kg +1

n
2

3
ε
(−3)

g


n

− 3 εg (−3)





0

m+kg +1
2

g g and g ∗( β ) + λ − µ = 1,
α ∈ F∗3 , β ∈ SR
α
α
β
g g and g ∗( ) + λ − µ = 2,
α ∈ F∗3 , β ∈ SR
α
α
otherwise.
12

Proof. It is clear that
S4 =

X

u∈F∗3

=

X

t∈F∗3

ζ3µu

X

X

ζ3λt σt

ζ3λt σt 

X

u∈F∗3

X

x∈F3n

y∈F3m

X
µu

X
T r n (x)+ u T r n (αx)

t∈F∗3



T r n (x)+ ut T r3n (αx)
ζ3 3

ζ3 t

ζ3 3

x∈F3n

t

3

g(y)+ ut T r3m (βy)
ζ3

g(y)+ ut T r3m (βy)

ζ3

y∈F3m





X
X
X
β
µ
m

g(y)−T r3 ( α y)
−
 3n
 α ∈ F∗3 ,
ζ3
ζ3 α
ζ3λt σt 
=
y∈F3m
u∈F∗3
t∈F∗3



0
α∈
/ F∗3 ,

µ
m+kg X
t(g ∗ ( β )+λ− α
)
m+k

3n εg (−3) 2
η0 g (t)ζ3 α
α ∈ F∗3 ,
=
t∈F∗p


0
α∈
/ F∗3 .

When m + kg is even, then we have

m+kg

n
2

2
·
3
ε
(−3)

g


m+kg
S4 =
n
2
−
3
ε
(−3)

g




0

!




β
µ
α ∈ F∗3 and g ∗ ( ) + λ − = 0,
α
α
β
µ
α ∈ F∗3 and g ∗ ( ) + λ − 6= 0,
α
α
otherwise.

When m + kg is odd, then we have

m+kg +1
β
µ

n
2

3
ε
(−3)
α ∈ F∗3 and g ∗( ) + λ − = 1,

g

α
α

m+kg +1
β
µ
S4 =
α ∈ F∗3 and g ∗( ) + λ − = 2,
− 3n εg (−3) 2


α
α



0
otherwise.
This completes the proof.

Lemma 3.8. Let notations be the same as Lemma 3.3. Let s = n + m, λ ∈ F∗3 , µ ∈ F3 and
X
XX
t2 (T r n (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)
ζ3 3
.
S5 =
t∈F∗3 u∈F∗3 (x,y)∈F3n ×F3m

For any (α, β) ∈ F3n × F3m \{(0, 0)}, the following statements hold.

m+kg

n
g g and g ∗( β ) + λ − µ = 0,
2

α ∈ F∗3 , β ∈ SR
2
·
3
ε
(−3)

g

α
α


√
g

β
( −3 − 1)3n ε (−3) m+k
g g and g ∗( ) + λ − µ = 1,
2
α ∈ F∗3 , β ∈ SR
g
S5 =
α
α

m+kg
√

β

g g and g ∗( ) + λ − µ = 2,

( −3 + 1)3n εg (−3) 2
α ∈ F∗3 , β ∈ SR


α
α



0
otherwise.
13

Proof. The proof is similar to that of Lemma 3.7 and we immediately have

β
µ
m+kg X
t2 (g ∗ ( α
)+λ− α
)
m+k

3n εg (−3) 2
η0 g (t2 )ζ3
α ∈ F∗3 ,
S5 =
t∈F∗3


0
α∈
/ F∗3 ,

m+kg
µ
β

2 · 3n εg (−3) 2

α ∈ F∗3 and g ∗( ) + λ − = 0,


α
α



 
m+kg
√
µ
µ
β
β
= 3n ε (−3) 2
−3 − 1
α ∈ F∗3 and g ∗( ) + λ − 6= 0,
η0 g ∗ ( ) + λ −
g


α
α
α
α



0
otherwise.

The desired result immediately then follows from the definition of η0 .
√
g g , where
e g (β) = εg 3∗ m+kg ζ g∗ (β) for every β ∈ SR
Lemma 3.9. Let g ∈ WRP with R
3
∗
εg ∈ {−1, 1} and 0 ≤ kg ≤ m. Let s = n + m, λ ∈ F3 , µ ∈ F3 and
Ng (0) = #{(x, y) ∈ F3n × F3m : T r3n (x) + g(y) + λ = 0, T r3s (αx + βy) + µ = 0}.

For any (α, β) ∈ F3n × F3m \{(0, 0)}, the following statements hold.
(1) If m + kg is even, then

m+kg

s−2
n−2
2

3
+
2
·
3
ε
(−3)

g


m+kg
Ng (0) =
s−2
n−2
2
3
−
3
ε
(−3)

g



 s−2
3

g g and g ∗ ( β ) + λ − µ = 0,
α ∈ F∗3 , β ∈ SR
α
α
β
g g and g ∗ ( ) + λ − µ 6= 0,
α ∈ F∗3 , β ∈ SR
α
α
otherwise.

(2) If m + kg is odd, then

m+kg +1

g g and g ∗ ( β ) + λ − µ = 1,

α ∈ F∗3 , β ∈ SR
3s−2 + 3n−2εg (−3) 2


α
α

m+kg +1
β
Ng (0) =
s−2
n−2
∗
g g and g ∗ ( ) + λ − µ = 2,
2
3
−
3
ε
(−3)
α
∈
F
,
β
∈
SR

g
3

α
α


 s−2
3
otherwise.

Proof. It is clear that
1 X X t(T r3n (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)
ζ
Ng (0) =
9 t∈F u∈F 3
3
3
X
X
1X
1
t(T r n (x)+g(y)+λ)
ζ3 3
+
=3s−2 +
9 t∈F∗
9 u∈F∗
3 (x,y)∈F3n ×F3m

+

1XX

9 t∈F∗ u∈F∗
3

X

X

3 (x,y)∈F3n ×F3m

t(T r3n (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)

ζ3

3 (x,y)∈F3n ×F3m

1
1
=3s−2 + S2 + S4 .
9
9
Then the desired results follow from Lemmas 3.5 and 3.7.
14

u(T r3s (αx+βy)+µ)

ζ3

4

Several new families of ternary self-orthogonal codes

In this section, we present several new explicit constructions of infinite families of ternary
self-orthogonal codes by using weakly regular plateaued functions (WRP). To this end,
we need the following useful result.
Lemma 4.1 ( [25]). Let C be any ternary linear code. Then C is self-orthogonal if every
codeword of C has weight which is divisible by 3.
Three new families of ternary self-orthogonal codes from f, g ∈ WRP

4.1

In this subsection, we consider the augmented code CDf g (0) defined in Equation (5) with
the defining set Df g (0) given by
Df g (0) = {(x, y) ∈ F3n × F3m : f (x) + g(y) + λ = 0} ,

(11)

where f, g ∈ WRP and λ ∈ F∗3 .
Let nf g (0) denote the length of the augmented code CDf g (0) and wt(cf g (0)) denote the
weight of nonzero codeword cg (0). Let s = n + m, λ ∈ F∗3 , µ ∈ F3 and
Nf g (0) = # {(x, y) ∈ F3n × F3m : f (x) + g(y) + λ = 0, T r3s (αx + βy) + µ = 0}

(12)

for any (α, β) ∈ F3n × F3m . Then it can be verified that
nf g (0) = # {(x, y) ∈ F3n × F3m : f (x) + g(y) + λ = 0}

(13)

and
wt(cf g (0)) =nf g (0) − Nf g (0)
!
X
1 X t(f (x)+g(y)+λ) 1 X X t(f (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)
=
ζ
ζ
−
3 t∈F 3
9 t∈F u∈F 3
(x,y)∈F3n ×F3m

3

3

3

2
1
1
=2 · 3s−2 + S1 − S2 − S3 ,
9
9
9

where
S1 =

X

X

t(f (x)+g(y)+λ)

ζ3

,

t∈F∗3 (x,y)∈F3n ×F3m

S2 =

X

X

u(T r3s (αx+βy)+µ)

ζ3

and

u∈F∗3 (x,y)∈F3n ×F3m

S3 =

XX

X

t(f (x)+g(y)+λ)+u(T r3s (αx+βy)+µ)

ζ3

t∈F∗3 u∈F∗3 (x,y)∈F3n ×F3m

15

.

Theorem 4.2. Let s = n + m, where n and m are two positive integers. Let f, g ∈
√
√
g f and R
e g (β) = εg 3∗ m+kg ζ3g∗ (β)
e f (α) = εf 3∗ n+kf ζ3f ∗ (α) for every α ∈ SR
WRP with R
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m. Assume that
for every β ∈ SR
s + kf + kg is even and kf + kg < s − 3, then CDf g (0) is a ternary five-weight self-orthogonal
[3s−1 + εf εg (−3)

s+kf +kg −2
2

, s + 1] code and its weight distribution is given in Table 1.

Proof. By the definition of nf g (0) in Equation (14) and Lemma 3.4, we have nf g (0) =
3s−1 + εf εg (−3)

s+kf +kg −2
2

when s + kf + kg is even. It clear the dimension of CDf g (0) is
s+kf +kg −2

2
, s + 1] code. It follows from the definition
s + 1, then this is a [3s−1 + εf εg (−3)
g f × SR
g g \{(0, 0)}, then
of wt(cf g (0)) and Lemmas 3.4, 3.5 and 3.6 that if (α, β) ∈
/ SR
s+kf +kg −2
g f × SR
g g S{(0, 0)}, then
2
wt(cf g (0)) = 2 · 3s−2 − 2εf εg (−3)
; and if (α, β) ∈ SR

wt(cf g (0)) =


0
µ = 0 and (α, β) = (0, 0),



s+kf +kg −2



2

µ = 0 and f ∗ (α) + g ∗(β) + λ = 0,
2 · 3s−2 + 2εf εg (−3)



s+kf +kg −2


2 · 3s−2 + εf εg (−3)
2
µ = 0, f ∗ (α) + g ∗ (β) + λ 6= 0, (α, β) 6= 0,





2 · 3s−2






s+kf +kg −2

 s−1
2
3 + εf εg (−3)

µ = 0, f ∗ (α) + g ∗ (β) + λ 6= 0, (α, β) 6= 0
or µ 6= 0, f ∗(α) + g ∗ (β) + λ = 0, (α, β) 6= 0,

µ 6= 0 and (α, β) = (0, 0).

The weight distribution of the code is straightly derived from Lemma 3.1 and 3.2.
s+kf +kg −2
Since s + kf + kg is even and kf + kg < s − 3, then s − 2 > 1 and
≥ 1 is
2
an integer. It implies that 3 | wt(cf g (0)) for any cf g (0) ∈ CDf g (0) . Then it turns out from
Lemma 4.1 that CDf g (0) is self-orthogonal and we complete the whole proof.

Table 1: The weight distribution of CDf g (0) when s + kf + kg is even.
Weight i
Multiplicity Ai
0
1
2 · 3s−2 + 2εf εg (−3)
2 · 3s−2 + εf εg (−3)
2 · 3s−2

3s−1 + εf εg (−3)

s+kf +kg −2
2

s+kf +kg −2
2

s+kf +kg −2
2

2 · 3s−2 − 2εf εg (−3)

s+kf +kg −2
2

3s−kf −kg −1 − (−1)s+1 εf εg (−3)

s−kf −kg −2
2

4 · 3s−kf −kg −1 + 2(−1)s+1 εf εg (−3)
4 · 3s−kf −kg −1 − (−1)s+1 εf εg (−3)

2

s−kf −kg −2
2

s−kf −kg −2
2

−2

−1

3s+1 − 3s−kf −kg +1

Example 4.3. Let n = 4 and m = 3, then s = n + m = 7. Let f (x) = T r34 (2x92 )
and g(x) = T r33 (ωx13 + ω 7 x4 + ω 7 x3 + ωx2 ), where ω is a primitive element of F33 with
16

ω 3 +2ω +1 = 0. On the one hand, it can be checked that f (x) is a ternary weakly regular 2e f (α) ∈ {0, −33 ζ f ∗ (α) } for all α ∈ F33 , where the sign of the Walsh
plateaued function with R
3
transform of f (x) is εf = 1 and f ∗ is an unbalanced function over F33 satisfying f ∗ (0) = 0.
On the other hand, it follows that g(x) is a ternary weakly regular 1-plateaued function and
the sign of the Walsh transform of g(x) is εg = −1. Therefore, it turns out from Theorem
4.4 that for λ ∈ F∗3 , CDf g (0) is a ternary four-weight self-orthogonal [648, 8, 486] code and
its weight enumerator is
1 + 24x324 + 112x405 + 104x486 + 6320x648 .
Verified by Magma [2], these results are true.
Theorem 4.4. Let s = n + m, where n and m are two positive integers. Let f, g ∈ WRP
√
√
g f and R
e g (β) = εg 3∗ m+kg ζ g∗ (β) for every
e f (α) = εf 3∗ n+kf ζ f ∗ (α) for every α ∈ SR
with R
3
3
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m. Assume that s + kf + kg is
β ∈ SR
odd and kf + kg < s − 3, then the following statements hold.
(1) If λ

=

εf εg (−3)
(2) If λ

=

εf εg (−3)

1,

then CDf g (0) is a ternary six-weight self-orthogonal [3s−1 −

s+kf +kg −1
2

, s + 1] code and its weight distribution is given in Table 2.

2,

then CDf g (0) is a ternary six-weight self-orthogonal [3s−1 +

s+kf +kg −1
2

, s + 1] code and its weight distribution is given in Table 3.

Proof. (1) Since λ = 1, then by the definition of nf g (0) in Equation (14) and Lemma 3.4,
we have nf g (0) = 3s−1 − εf εg (−3)

s+kf +kg −1
2

when s + kf + kg is odd. It clear the dimension
s+kf +kg −1

s−1

2
of CDf g (0) is s + 1, then this is a [3 − εf εg (−3)
, s + 1] code. It follows from the
g f × SR
g g \{(0, 0)},
definition of wt(cf g (0)) and Lemmas 3.4, 3.5 and 3.6 that if (α, β) ∈
/ SR
s+kf +kg −3
S
g f × SR
g g {(0, 0)}, then
2
then wt(cf g (0)) = 2 · 3s−2 + 2εf εg (−3)
; and if (α, β) ∈ SR


0




s+kf +kg −3

s−2

2

2
·
3
+
2ε
ε
(−3)
f g





2 · 3s−2



s+kf +kg −3

s−2
2
wt(cf g (0)) = 2 · 3 + 4εf εg (−3)

s+kf +kg −1


s−2
2

2
·
3
−
ε
ε
(−3)

f
g



s+kf +kg −3


2

2 · 3s−2 + εf εg (−3)



s+kf +kg −1

 s−1
2
3 − εf εg (−3)

µ = 0 and (α, β) = (0, 0),

(α, β) 6= (0, 0) and f ∗ (α) + g ∗ (β) + λ = 0,
µ = 0, (α, β) 6= (0, 0), f ∗(α) + g ∗ (β) + λ = 1,
µ = 0, (α, β) 6= (0, 0), f ∗(α) + g ∗ (β) + λ = 2,
µ 6= 0, (α, β) 6= (0, 0), f ∗(α) + g ∗ (β) + λ = 1,
µ 6= 0, (α, β) 6= (0, 0), f ∗(α) + g ∗ (β) + λ = 2,
µ 6= 0 and (α, β) = (0, 0).

(2) Since λ = 2, then by the definition of nf g (0) in Equation (14) and Lemma 3.4, we

have nf g (0) = 3s−1 + εf εg (−3)

s+kf +kg −1
2

when s + kf + kg is odd. It clear the dimension of

CDf g (0) is s + 1, then this is a [3s−1 + εf εg (−3)

s+kf +kg −1
2

17

, s + 1] code. It again follows from the

g f × SR
g g \{(0, 0)},
definition of wt(cf g (0)) and Lemmas 3.4, 3.5 and 3.6 that if (α, β) ∈
/ SR
s+kf +kg −3
S
g f × SR
g g {(0, 0)}, then
2
; and if (α, β) ∈ SR
then wt(cf g (0)) = 2 · 3s−2 − 2εf εg (−3)

0




s+kf +kg −3


2

2 · 3s−2 − 2εf εg (−3)



s+kf +kg −3


s−2

2

2
·
3
−
4ε
ε
(−3)
f
g


s−2
wt(cf g (0)) = 2 · 3

s+kf +kg −3


s−2
2

2
·
3
−
ε
ε
(−3)

f g



s+kf +kg −1


2

2 · 3s−2 + εf εg (−3)



s+k
+k
−1

g
f
 s−1
2
3 + εf εg (−3)

µ = 0 and (α, β) = (0, 0),

(α, β) 6= (0, 0) and f ∗ (α) + g ∗(β) + λ = 0,
µ = 0, (α, β) 6= (0, 0), f ∗(α) + g ∗(β) + λ = 1,
µ = 0, (α, β) 6= (0, 0), f ∗(α) + g ∗(β) + λ = 2,
µ 6= 0, (α, β) 6= (0, 0), f ∗(α) + g ∗(β) + λ = 1,
µ 6= 0, (α, β) 6= (0, 0), f ∗(α) + g ∗(β) + λ = 2,
µ 6= 0 and (α, β) = (0, 0).

The weight distributions of the codes are also straightly derived from Lemma 3.1 and 3.2.
s+kf +kg −3
≥ 1 is an
Since s + kf + kg is odd and kf + kg < s − 3, then s − 2 > 1 and
2
integer. It implies that 3 | wt(cf g (0)) for any c ∈ CDf g (0) . Then it turns out from Lemma
4.1 that CDf g (0) is self-orthogonal and we complete the proof.

Remark 4.5. Note that if we take m = kf = kg = 0 and εg = 1, then Theorem 4.2 can yield
n−2
an infinite family of ternary self-orthogonal [3n−1 + εf (−3) 2 , n + 1] codes, which coincide
with [21, Theorem 1]. Hence, Theorem 4.2 can be seen as a generalization of [21] and it
can produce ternary self-orthogonal codes with more flexible parameters. In addition, it
is easy to see that Theorem 4.4 can also similarly yield new infinite families of ternary
self-orthogonal codes with flexible parameters.

Table 2: The weight distribution of CDf g (0) when kf + kg is odd and λ = 1.
Weight i
Multiplicity Ai
0
1
2 · 3s−2 + 2εf εg (−3)
2 · 3s−2

2 · 3s−2 + 4εf εg (−3)

2 · 3s−2 − εf εg (−3)
2 · 3s−2 + εf εg (−3)

3s−1 − εf εg (−3)

s+kf +kg −3
2

s+kf +kg −3
2

s+kf +kg −1
2

s+kf +kg −3
2

s+kf +kg −1
2

3s+1 − 2 · 3s−kf −kg + (−1)s εf εg (−3)
3s−kf −kg −1 − 1
3s−kf −kg −1 + (−1)s εf εg (−3)
2 · 3s−kf −kg −1 − 2

s−kf −kg −1
2

2 · 3s−kf −kg −1 + 2(−1)s εf εg (−3)
2

18

s−kf −kg +1
2

s−kf −kg −1
2

Table 3: The weight distribution of CDf g (0) when kf + kg is odd and λ = 2.
Weight i
Multiplicity Ai
0
1
2 · 3s−2 − 2εf εg (−3)

2 · 3s−2 − 4εf εg (−3)
2 · 3s−2
2 · 3s−2 − εf εg (−3)

2 · 3s−2 + εf εg (−3)
3s−1 + εf εg (−3)

s+kf +kg −3
2
s+kf +kg −3
2

s+kf +kg −3
2

s+kf +kg −1
2

s+kf +kg −1
2

3s+1 − 2 · 3s−kf −kg − (−1)s εf εg (−3)
3s−kf −kg −1 − (−1)s εf εg (−3)
3s−kf −kg −1 − 1

s−kf −kg +1
2

s−kf −kg −1
2

2 · 3s−kf −kg −1 − 2(−1)s εf εg (−3)

s−kf −kg −1
2

2 · 3s−kf −kg −1 − 2

2

Example 4.6. Let n = 4 and m = 3, then s = n + m = 7. Let f (x) = T r34 (2x92 ) and
g(x) = T r33 (ξx4 ), where ξ is a primitive element of F33 . On the one hand, it can be checked
e f (α) ∈ {0, −33 ζ3f ∗ (α) } for
that f (x) is a ternary weakly regular 2-plateaued function with R
all α ∈ F34 , where the sign of the Walsh transform of f (x) is εf = 1 and f ∗ is an unbalanced
function over F33 satisfying f ∗ (0) = 0. On the other hand, it follows that g(x) is a ternary
weakly regular 0-plateaued and the sign of the Walsh transform of g(x) is εg = −1. Hence,
it turns out from Theorem 4.4 that for λ = 1, CDf g (0) is a ternary six-weight self-orthogonal
[810,8,486] code and its weight enumerator is
1 + 80x486 + 180x513 + 6048x540 + 160x567 + 90x594 + 2x810 .
Verified by Magma [2], these results are true.
Example 4.7. Let n = 3 and m = 3, then s = m + n = 6. Let f (x) = T r33 (ξ 22 x13 +
ξ 7 x4 + ξx2 ), where ξ is a primitive element of F33 , then it follows that f (x) is a ternary
e f (α) ∈ {0, 9εf ζ3f ∗ (α) } = {0, −9, −9ζ3 , −9ζ 2}
weakly regular 1-plateaued function with R
3
for all α ∈ F33 , where the sign of the Walsh transform of f (x) is εf = −1 and f ∗ is an
unbalanced function over F33 satisfying f ∗ (0) = 0. Let g(x) = T r33 (ωx2 ), where ω is a
primitive element of F33 , then it can be checked that g(x) is a ternary weakly regular 0plateaued and the sign of the Walsh transform of g(x) is εg = −1. Hence, it turns out from
Theorem 4.4 that for λ = 2, CDf g (0) is a ternary six-weight self-orthogonal [216,7,126] code
and its weight enumerator is
1 + 72x126 + 160x135 + 1728x144 + 144x153 + 80x162 + 2x216 .
Verified by Magma [2], these results are true.
4.2

Two new families of ternary self-orthogonal codes from f ∈
/ WRP and
g ∈ WRP.

In this subsection, we let notations be the same as above and assume that f (x) = T r3n (x)
and g(y) ∈ WRP. Let s = n + m and denote by
Dg (0) = {(x, y) ∈ F3n × F3m : T r3n (x) + g(y) + λ = 0} ,
19

where λ ∈ F∗3 . Let ng (0) denote the length of the augmented codes CDg (0) and wt(cg (0))
denote the weight of nonzero codewords cg (0). It can be verified that
ng (0) = # {(x, y) ∈ F3n × F3m : T r3n (x) + g(y) + λ = 0}

(14)

and then
ng (0) =

1X
3 t∈F

X

t(T r3n (x)+g(y)+λ)

ζ3

= 3s−1 .

3 (x,y)∈F3n ×F3m

Then we determine their weight distributions in the following theorems.
√
e g (β) = εg 3∗ m+kg ζ g∗ (β) for every β ∈ SR
g g , where
Theorem 4.8. Let g ∈ WRP with R
3
∗
εg ∈ {−1, 1} and 0 ≤ kg ≤ m. If m + kg is even and λ ∈ F3 , then CDg (0) is a four-weight
self-orthogonal [3s−1 , s + 1] ternary linear code with weight distribution given in Table 4.
Proof. The proof is very similar to that of Theorem 4.2 and the main difference is that we
use Lemmas 3.8 and 3.9 here.
√
g g , where
e g (β) = εg 3∗ m+kg ζpg∗ (β) for every β ∈ SR
Theorem 4.9. Let g ∈ WRP with R
∗
εg ∈ {−1, 1} and 0 ≤ kg ≤ m. If m + kg is odd and λ ∈ F3 , then CDg (0) is a four-weight
self-orthogonal [3s−1 , s + 1] ternary linear code with weight distribution given in Table 5.
Proof. The proof is very similar to that of Theorem 4.8 and we omit it here.

Table 4: The weight distribution of CDg (0) when m + kg is even.
Weight i
Multiplicity Ai
0
1
2 · 3s−2 − 2 · 3n−2 εg (−3)

2 · 3s−2 + 3n−2 εg (−3)
2 · 3s−2
3s−1

m+kg
2

m+kg
2

2 · 3m−kg

4 · 3m−kg
3s+1 − 2 · 3m−kg +1 − 3
2

Table 5: The weight distribution of CDg (0) when m + kg is odd.
Weight i
Multiplicity Ai
0
1
2 · 3s−2 − 3n−2εg (−3)

2 · 3s−2 + 3n−2 εg (−3)
2 · 3s−2
3s−1

m+kg +1
2
m+kg +1
2

20

2 · 3m−kg

2 · 3m−kg
3s+1 − 4 · 3m−kg − 3
2

Example 4.10. Let n = 1 and m = 4, then s = n + m = 5. Let g(x) = T r34(ξx2 ), where
ξ is a primitive element of F34 , it follows that g(x) is a ternary weakly regular 0-plateaued
function and the sign of the Walsh transform of g(x) is εg = −1. Therefore, it turns out
from Theorem 4.8 that CDg (0) is a ternary four-weight self-orthogonal [81,6,51] code and its
weight enumerator is
1 + 324x51 + 240x54 + 162x60 + 2x81 ,
which is optimal according to [17]. Verified by Magma [2], these results are true.
Example 4.11. Let n = 1 and m = 3, then s = n + m = 4. Let g(x) = T r33(ξx4 ), where
ξ is a primitive element of F33 , it follows that g(x) is a ternary weakly regular 0-plateaued
function and the sign of the Walsh transform of g(x) is εg = −1. Therefore, it turns out
from Theorem 4.9 that CDg (1) is a ternary four-weight self-orthogonal [27,5,15] code and its
weight enumerator is
1 + 54x15 + 132x18 + 54x21 + 2x27 ,
which is almost optimal since the optimal code with length 27 and dimension 5 has minimum
weight 16 by [17]. Verified by Magma [2], these results are true.

5

Dual codes of several families of ternary self-orthogonal codes

In this section, we study dual codes of these ternary self-orthogonal linear codes con⊥
⊥
structed in Section 4. Let CDf g (0) and CDg (0) denote the dual codes of CDf g (0) and CDg (0) ,
respectively. Next, we determine their parameters.
Theorem 5.1. Let s = n +∗ m, where n and m are two positive integers. Let ∗f, g ∈ WRP
e f (α) = εf √p∗ n+kf ζpf (α) for every α ∈ SR
g f and R
e g (β) = εg √p∗ m+kg ζpg (β) for every
with R
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m. Then the following
β ∈ SR
statements hold.
(1) If s + kf + kg is even, kf + kg < s − 3 and λ ∈ F∗3 , then CDf g (0)
[3s−1 + εf εg (−3)

s+kf +kg −2
2

, 3s−1 + εf εg (−3)

s+kf +kg −2
2

s+kf +kg −1
2

, 3s−1 − εf εg (−3)

s+kf +kg −1
2

s+kf +kg −1
2

, 3s−1 + εf εg (−3)

s+kf +kg −1
2

⊥

is a ternary [3s−1 −

− s − 1, 3] linear code.

(3) If s + kf + kg is odd, kf + kg < s − 3 and λ = 2, then CDf g (0)
εf εg (−3)

is a ternary

− s − 1, 3] linear code.

(2) If s + kf + kg is odd, kf + kg < s − 3 and λ = 1, then CDf g (0)
εf εg (−3)

⊥

⊥

is a ternary [3s−1 +

− s − 1, 3] linear code.

⊥

⊥

Proof. (1) Let d(CDf g (0) ) denote the minimum weight of CDf g (0) . It follows from the
⊥

⊥
⊥
definition of CDf g (0) that d(CDf g (0) ) ≥ 2. Let (1, A1 , A2 , · · · , An ) and (1, A⊥
1 , A2 , · · · , An )
⊥

denote the weight distributions of CDf g (0) and CDf g (0) , respectively. If s + kf + kg is even,
21

it follows from the third Pless power moment (P3 ) that
n
X
j=0



⊥
j 2 Aj = 3k−2 2n(3n − n + 1) − (6n − 3 − 2n + 2)A⊥
1 + 2A2 .

(15)

Substituting the weight distribution of CDf g (0) we obtain in Theorem 4.2 into Equation (15),
we immediately have
A⊥
2 = 0.
Note also that the fourth Pless power moment (P4 ) yields that
n
X
j=0

j 3 Aj = 3k−3 [2n(32 n2 − 6n2 + 9n − 3 + n2 − 3n + 2) − 6A⊥
3 ].

(16)

By combining the weight distribution of CDf g (0) we present in Theorem 4.2 with Equation
(16), we get
s+kf +kg −2


2s−2
s−2
2s−kf −kg −3
−1
2
A⊥
=
10
·
3
−
16
·
3
−
32
·
3
−
3
(−3)
3

− 88 · 33s−kf −kg −6 − 4 · 32s+2 − 8 · 32s−kf −kg −3 + 20 · 32s−5 > 0
⊥

and hence, d(CDf g (0) ) = 3. This completes the desired result (1).
(2) and (3) By taking an argument completely similar to that of (1) above, the desired
results (2) and (3) hold.
Theorem 5.2. Let s = n + m, where n and m are two positive integers. Let λ ∈ F∗3 and
g g , where εg ∈ {−1, 1} and
e g (β) = εg √p∗ m+kg ζpg∗ (β) for every β ∈ SR
g ∈ WRP with R
0 ≤ kg ≤ m. Then the following statements hold.
⊥

(1) If m + kg is even, then CDg (0) is a ternary [3s−1 , 3s−1 − s − 1, 3] linear code.
⊥

(2) If m + kg is odd, then CDg (0) is a ternary [3s−1 , 3s−1 − s − 1, 3] linear code.
⊥

Proof. Let d⊥ denote the minimum weight of CDg (0) and it follows from the definition of
⊥
⊥
⊥
CDg (0) that d⊥ ≥ 2. Let (1, A1 , A2 , · · · , An ) and (1, A⊥
1 , A2 , · · · , An ) denote the weight
⊥

distributions of CDg (0) and CDg (0) , respectively.
For even m + kg , it follows from the third Pless power moment (P3 ) that
n
X
j=0



⊥
j 2 Aj = 3k−2 2n(3n − n + 1) − (6n − 3 − 2n + 2)A⊥
+
2A
1
2 .

(17)

Substituting the weight distribution of CDg (0) we obtain in Theorem 4.2 into Equation (17),
we have
A⊥
2 = 0.
22

From the fourth Pless power moment, we also derive
n
X
j=0

j 3 Aj = 3k−3 [2n(32 n2 − 6n2 + 9n − 3 + n2 − 3n + 2) − 6A⊥
3 ].

(18)

Substituting the weight distribution of CDg (0) we obtain in Theorem 4.2 into Equation (18),
then we obtain




m+kg
⊥
s−2
n−2
m
2
A3 = 3
3
2 · εg (−3)
+ 3 − 1 > 0,
which further deduces that d⊥ = 3.
For odd m + kg , it follows from Theorem 4.9 and Equation (17) that
A⊥
2 = 0.
With Theorem 4.9 and Equation (18) again, we have



s−2
A⊥
8 · 32s−2 + 3s−2 − 1 > 0
3 = 3

and hence, d⊥ = 3. This completes the proof.

⊥

Remark 5.3. Although the dual codes CDg (0) in Parts (1) and (2) of Theorem 5.2 have
the same parameters, they must be not equivalent since their original codes CDg (0) have
different weight distributions according to Tables 4 and Table 5.

6

Application to ternary LCD codes

In this section, we apply our new ternary self-orthogonal codes constructed in Section
4 to obtain new families of ternary LCD codes. To this end, we need to recall some useful
definitions and results as follows.
Let I be an identity matrix and O be a zero matrix. For any matrix G, if GGT = I, we
call G a row-orthogonal matrix; and if GGT = O, we call G a row-self-orthogonal matrix.
A leading-systematic linear code is referred as a linear code generated by a matrix of the
form G = (I A). Moreover, the matrix G is called the systematic generator matrix of the
code. The relationship between LCD codes and row-orthogonal matrices has been proposed
in [39].
Lemma 6.1 ( [39]). A leading-systematic linear code C is an LCD code if its systematic
generator matrix G = (I A) is row-orthogonal, i.e., the matrix A is row-self orthogonal.
Lemma 6.2. Let p be an odd prime and ξ be a generator element of F∗ps = hξi. For any
p-ary linear augmented code CD defined in Equation (4) with the defining set D, where
D = {d1 , d2 , · · · , dt }, a generator matrix of CD is given by


T rps (d1 )
T rps (d2 )
···
T rps (dt )
 T r s (ξd1)
T rps (ξd2)
· · · T rps (ξdt ) 
p




.
.
.
.
.
.
.
.
(19)
G=
.
.
.
.
.


 T rps (ξ s−1 d1 ) T rps (ξ s−1d2 ) · · · T rps (ξ s−1dt ) 
1
1
···
1
23

Proof. The generator matrix G follows from the definition of the augmented code CD in
Equation (4) and the fact that {1, ξ, · · · , ξ s−1} forms a basis of Fps over Fp .
Theorem 6.3. Let s = n + m, where n and m are two positive integers. Let f, g ∈ WRP
√
√
g f and R
e g (β) = εg 3∗ m+kg ζ3g∗ (β) for every
e f (α) = εf 3∗ n+kf ζ3f ∗ (α) for every α ∈ SR
with R
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m. Assume that s + kf + kg
β ∈ SR
is even and kf + kg < s − 3. Let λ ∈ F∗3 and CDf g (0) be defined in Equation (4), then
s−1
the matrix G = (I G) generates
n a ternary LCD [3 + εf εg (−3)

s+kf +kg −2
2

LCD code C, where d ≥ min 1 + 2 · 3s−2, 1 + 2 · 3s−2 + 2εf εg (−3)

+os + 1, s + 1, d]
and G is the

s+kf +kg −2
2

generator matrix of CDf g (0) . Besides, C ⊥ is a ternary LCD [3s−1 + εf εg (−3)

s+kf +kg −2
2

+

s+kf +kg −2

2
s + 1, 3s−1 + εf εg (−3)
, 3] code and it is at least almost optimal with respect to the
Sphere-packing bound given in Equation (1).

Proof. According to Theorem 4.2 and Lemma 6.1, the matrix G = (I G) generates a
ternary LCD code C, where


1 0 ··· 0 0
T r3s (d1 )
T r3s (d2 )
···
T r3s (dt )
 0 1 · · · 0 0 T r s (ξd1 )
T r3s (ξd2) · · · T r3s (ξdt) 
3


 .. .. . . .. ..

.
.
.
.
.
.
.
.
G= . .
(20)

. . .
.
.
.
.


s
s−1
s
s−1
s
s−1
 1 0 · · · 1 0 T r3 (ξ d1 ) T r3 (ξ d2 ) · · · T r3 (ξ dt ) 
1 0 ··· 0 1
1
1
···
1
and Df g (0) = {d1 , d2 , · · · , dt }. Note also that a dual code of an LCD code is still LCD.
Hence, C ⊥ is also a ternary LCD code. Let d(CDf g (0) ) denote the minimum weight of CDf g (0)
and d denote the minimum weight of C. Then it is clear that d ≥ d(CDf g (0) ) + 1. Moreover,
o
n
s+kf +kg −2
s−2
s−2
2
.
it turns out from Table 1 that d ≥ min 1 + 2 · 3 , 1 + 2 · 3 + 2εf εg (−3)
⊥

By Theorem 5.1, the minimum weight of the dual code CDf g (0) is 3, which implies that
any two columns of G are linearly independent and there exist three dependent columns of G
over F3 . Let d⊥ denote the minimum weight of C ⊥ . Now, we prove that d⊥ = 3. Then it suffices to prove that the column vector (0, 0, · · · , 0, 1)T and any column of G are linearly independent over F3 . Suppose that (0, 0, · · · , 0, 1)T and (T r3s (di ), T r3s (ξdi ), · · · , T r3s(ξ s−1 di ), 1)T
are linearly dependent over F3 for some di ∈ Df g (0), then we have


T r3s (di ) = 0,


 T r s (ξdi ) = 0,
3
..

.


 T r s (ξ s−1 d ) = 0.
i
3
For any x =

Ps−1

i−0 ki ξ

i

∈ F3s and ki ∈ F3 , it then deduces that
T r3s (xdi ) =

s−1
X

ki T r3s (ξ i di ) = 0.

i=0

24

/ Df g (0)
Note that T r3s (x) is a 3-ary function from F3s to F3 and |Ker(T r3s )| = 3s−1 since 0 ∈
for any λ ∈ F∗3 . This produces a contradiction and hence, any two columns of G are linearly
independent over F3 . That is d⊥ = 3.
s+kf +kg −2
2
Moreover, for any fixed length 3s−1 + εf εg (−3)
+ s + 1 and dimension 3s−1 +
s+kf +kg −2

2
εf εg (−3)
, it follows from the Sphere-packing bound given in Equation (1) that
⊥
d is at most 4. Hence, C ⊥ is at least almost optimal with respect to the sphere-packing
bound.
We have completed the whole proof.

Theorem 6.4. Let s = n + m, where n and m are two positive integers. Let f, g ∈ WRP
√
√
g f and R
e g (β) = εg 3∗ m+kg ζ g∗ (β) for every
e f (α) = εf 3∗ n+kf ζ f ∗ (α) for every α ∈ SR
with R
3
3
g g , where εf , εg ∈ {−1, 1}, 0 ≤ kf ≤ n and 0 ≤ kg ≤ m. Assume that s + kf + kg is
β ∈ SR
odd, kf + kg < s − 3 and λ ∈ F∗3 , CDf g (0) be defined in Equation (4) with a generator matrix
G. Then the following statements hold.
(1) If λ

=

1,

then the matrix G

=

(I G) generates a ternary LCD

s+kf +kg −1
2

s−1

+ s + 1, s + 1,
n − εf εg (−3)
o d]
s+kf +kg −3
2
min 1 + 2 · 3s−2 , 1 + 2 · 3s−2 + 4εf εg (−3)
.

[3

s+kf +kg −1

code

C,

where

d

≥

Besides, C ⊥ is a ternary

s+kf +kg −1

2
2
+ s + 1, 3s−1 − εf εg (−3)
, 3] code and it is at
LCD [3s−1 − εf εg (−3)
least almost optimal with respect to the sphere-packing bound given in Equation (1).

(2) If λ

=

2,

then the matrix G

[3s−1n + εf εg (−3)

s+kf +kg −1
2

=

(I G) generates a ternary LCD

+ s + 1, s + 1,
od]
s+kf +kg −3
2
.
min 1 + 2 · 3s−2 , 1 + 2 · 3s−2 − 4εf εg (−3)
s+kf +kg −1

code

C,

where

d

≥

Besides, C ⊥ is a ternary

s+kf +kg −1

2
2
LCD [3s−1 + εf εg (−3)
+ s + 1, 3s−1 + εf εg (−3)
, 3] code and it is at
least almost optimal with respect to the sphere-packing bound given in Equation (1).

Proof. The proof is very similar to that of Theorem 6.3 and the main difference is that we
use Theorems 4.2, 4.4, 5.1 and Lemma 6.1 here.
Theorem 6.5. Let s = n + m, where n and m are two positive integers. Let g ∈ WRP
√
e g (β) = εg 3∗ m+kg ζ3g∗ (β) for β ∈ SR
g g , where εg ∈ {−1, 1} and 0 ≤ kg ≤ m. Let
with R
codes CDg (0) be defined in Equation (4) and G be the generator matrix of CDg (0) . Let λ ∈ F∗3 ,
then we have the following.
=
(I G) generates
(1) If m + kg be even, then the matrix G
s−1
a ternary
LCD [3
+ s + 1, s + 1, d] code C, where
d
≥
n
o
m+kg
m+kg
s−2
n−2
s−2
n−2
min 1 + 2 · 3 − 2 · 3 εg (−3) 2 , 1 + 2 · 3 + 3 εg (−3) 2
.
Besides,
C ⊥ is a ternary LCD [3s−1 + s + 1, 3s−1 , 3] code which is at least almost optimal with
respect to the sphere-packing bound given in Equation (1).

(2) If
a

m + kg be
ternary LCD

odd, then the matrix G
=
[3s−1 + s + 1, s + 1, d] code
25

(I G) generates
C, where d
≥

o
n
m+kg
m+kg
.
min 1 + 2 · 3s−2 − 3n−2 εg (−3) 2 , 1 + 2 · 3s−2 + 3n−2 εg (−3) 2

Besides,

C ⊥ is a ternary LCD [3s−1 + s + 1, 3s−1 , 3] code which is at least almost optimal with
respect to the sphere-packing bound given in Equation (1).

Proof. The proof is similar to Theorem 6.3 and we obtain the desired results from Theorems
4.8, 4.9, 5.2 and Lemma 6.1.
Example 6.6. Let G denote the generator matrix of code CDf g (0) in Example 4.7. According to Theorem 6.3, (I G) generates a ternary LCD [223, 7, 127] code C and C ⊥ is a
ternary LCD [223, 216, 3] code. Note also that C ⊥ is optimal according to [17]. Verified by
Magma [2], these results are true.
Example 6.7. Let G denote the generator matrix of code CDg (0) in Example 4.10. According to Theorem 6.5, (I G) generates a ternary LCD [87,6,52] code C and C ⊥ is a optimal
ternary LCD [87,81,3] code. Note also that C ⊥ is optimal according to [17]. Verified by
Magma [2], these results are true.
Example 6.8. Let G denote the generator matrix of code CDg (0) in Example 4.11. According to Theorem 6.5, (I G) generates a ternary [32,5,16] LCD code C and C ⊥ is a optimal
ternary LCD [32,27,3] code. Note also that C ⊥ is optimal according to [17]. Verified by
Magma [2], these results are true.

7

Conclusions

In this paper, we constructed several new infinite families of ternary self-orthogonal
augmented codes with flexible parameters from weakly regular plateaued functions and we
completely determine their parameters and weight distributions as well as the parameters
of their dual codes. It is also worth noting that these families of ternary self-orthogonal
codes contain some (almost) optimal codes. As an application, we further derive several
new infinite families of ternary LCD codes from these self-orthogonal codes and some of
them are at least almost optimal according to the Sphere-packing bound given in Equation
(1). For future research, it would be interesting to construct more infinite families of p-ary
self-orthogonal codes with good parameters by using other constructions and functions.

Acknowledgments
This research is supported by the National Natural Science Foundation of China under
Grant No. U21A20428 and 12171134.
Data availability No data.
Conflict of Interest The authors declare that there is no possible conflict of interest.
26

Acknowledgments
This research is supported by the National Natural Science Foundation of China under
Grant No. U21A20428 and 12171134.

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30