Several families of q -ary minimal linear codes with wmin∕wmax≤(q−1)∕q

Discrete Mathematics 343 (2020) 111840

Contents lists available at ScienceDirect

Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

Several families of q-ary minimal linear codes with wmin /wmax ≤ (q − 1)/q Zexia Shi, Fang-Wei Fu Chern Institute of Mathematics, Nankai University, Tianjin 300071, PR China

article

info

Article history: Received 29 May 2019 Received in revised form 22 November 2019 Accepted 20 January 2020 Available online 7 February 2020 Keywords: Linear code Minimal code Weight distribution Secret sharing

a b s t r a c t Constructing minimal linear codes is an interesting research topic due to their applications in coding theory and cryptography. Ashikhmin and Barg pointed out that wmin /wmax > (q − 1)/q is a sufficient condition for a linear code over the finite field Fq to be minimal, where wmin and wmax respectively denote the minimum and maximum nonzero weights in a code. However, only a few families of minimal linear codes over Fq with wmin /wmax ≤ (q − 1)/q were reported in the literature. In this paper, we obtain several families of minimal q-ary linear codes with wmin /wmax ≤ (q − 1)/q. The weight distributions of all the constructed minimal linear codes are presented. © 2020 Elsevier B.V. All rights reserved.

1. Introduction Let q be a prime power and Fq be a finite field with q elements. An [n, k, d] linear code C over Fq is a k-dimensional subspace of Fnq with minimum Hamming distance d. Let Ai denote the number of codewords with Hamming weight i in C . Then (1, A1 , A2 , . . . , An ) is the weight distribution of C and 1 + A1 z + A2 z 2 + · · · + An z n is called the weight enumerator of C . For a vector v = (v1 , v2 , . . . , vn ) ∈ Fnq , let Suppt(v) denote the support of v, which is defined by Suppt(v) = {1 ≤ i ≤ n : vi ̸ = 0}. We call v the characteristic vector or the incidence vector of the set Suppt(v). For two vectors u, v ∈ Fnq , if Suppt(u) contains Suppt(v), then we say that u covers v. We write v ⪯ u if u covers v, and v ≺ u if Suppt(v) is a proper subset of Suppt(u). A codeword u of a linear code C is said to be minimal if u covers only the codeword au for all a ∈ Fq , but no other codewords of C . A linear code C is said to be minimal if every codeword of C is minimal. Constructing minimal linear codes is an interesting research topic due to their applications in coding theory and cryptography. For example, minimal linear codes could be decoded with a minimum distance decoding method [1]. They also have interesting applications in secret sharing [5,12,14] and secure two-party computations [6]. Ashikhmin and Barg [1] gave the following sufficient condition for a linear code to be minimal. Lemma 1 ([1] (Ashikhmin–Barg)). A linear code C over Fq is minimal if q−1 wmin > , wmax q where wmin and wmax denote the minimum and maximum nonzero Hamming weights in C , respectively. E-mail address: [email protected] (Z. Shi). https://doi.org/10.1016/j.disc.2020.111840 0012-365X/© 2020 Elsevier B.V. All rights reserved.

2

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

According to Ashikhmin–Barg’s condition, many minimal linear codes with wmin /wmax > (q − 1)/q have been reported in the literature (see for instance [5,8,13,14]). Recently, an infinite family of minimal binary linear codes violating Ashikhmin–Barg’s condition was presented in [7]. Later, Ding et al. [9] derived a necessary and sufficient condition for a binary linear code to be minimal and obtained three families of minimal binary linear codes violating Ashikhmin–Barg’s condition from a generic construction. In [10], Heng et al. presented a necessary and sufficient condition on minimal linear codes over finite fields, which generalized the result of the binary case given in [9]. Moreover, a family of ternary minimal linear codes with wmin /wmax < 2/3 was constructed. In [2], Bartoli et al. provided families of minimal codes violating Ashikhmin–Barg’s condition by generalizing the constructions in [9,10] to the finite field of odd characteristic. Zhang et al. [15] obtained four families of minimal binary linear codes with wmin /wmax ≤ 1/2 from the linear codes proposed by Zhou et al. [16]. Very recently, Bonini et al. [3] investigated the link between minimal linear codes and blocking sets. Moreover, they presented an infinite family of minimal linear codes not satisfying the Ashikhmin–Barg’s condition. Inspired by [15], in this paper we obtain several families of minimal q-ary linear codes violating Ashikhmin–Barg’s condition by generalizing the linear codes proposed by Zhou et al. [16]. We first obtain two families of q-ary minimal linear codes with wmin /wmax ≤ (q − 1)/q from a generic construction by choosing two specific defining sets. By adding or puncturing the elements in the defining set, we obtain more minimal q-ary linear codes violating Ashikhmin–Barg’s condition. The weight distributions of all the constructed minimal linear codes are established. The rest of this paper is organized as follows. In Section 2, we recall some basic concepts and properties on character, and introduce a generic construction of linear codes. In Sections 3 and 4, we obtain two families of q-ary minimal linear codes violating Ashikhmin–Barg’s condition from a generic construction by choosing two specific defining sets. In Section 5, more families of minimal q-ary linear codes violating Ashikhmin–Barg’s condition are presented by adding or puncturing elements of the defining sets given in Sections 3 and 4. 2. Preliminaries Let q be a power of prime p. For b ∈ Fq , an additive character of Fq can be defined as follows

χb : Fq → C∗ , χb (x) = ζpTr(bx) , 2π

√ −1

where ζp = e p is a primitive pth root of unity and Tr denotes the trace function from Fq onto Fp . When b = 0, χ0 (x) = 1 for all x ∈ Fq , and is called the trivial additive character of Fq . When b = 1, we call χ1 the canonical additive character of Fq . It is not hard to see that χb (x) = χ1 (bx) for all b, x ∈ Fq . The orthogonal property of additive characters is given by [11]

∑ x∈Fq

χb (x) =

{

q,

if b = 0,

0,

if b ∈ F∗q .

Let m be a positive integer. Ding [8] proposed a generic construction of linear codes over Fp from subsets of Fq . In [16], Zhou et al. restated this generic construction from the view point of vector space. A general version of this construction is presented in the following. Let D = {g1 , g2 , . . . , gn } ⊆ Fm q . Using D, we can define a linear code of length n over Fq by CD = {(a · g1 , a · g2 , . . . , a · gn ) : a ∈ Fm q }.

(1)

The set D is called the defining set of the code CD . Let G be the m × n matrix defined by G = [g1 g2 . . . gn ].

(2)

It is not hard to see that CD is the linear code generated by the row vectors of G. Denote Rank(G) the rank of the matrix G. Thus CD is an [n, k] linear code over Fq with k = Rank(G). In particular, G is exactly a generator matrix of CD when Rank(G) = m. 3. The first family of minimal linear codes Let q be a prime power and m be a positive integer. In this section, based on a generic construction given in (1), we construct a family of minimal linear codes over Fq violating Ashikhmin–Barg’s condition by choosing a specific defining set D. Let D1,2 be the set of nonzero vectors in Fm q with weight no more than two, i.e. D1,2 = {g ∈ Fm q : wt(g) = 1 or wt(g) = 2}. Based on the general construction given in (1), we have the linear code CD1,2 = {(a · g1 , a · g2 , . . . , a · gn ) : a ∈ Fm q },

where n = |D1,2 | and D1,2 = {g1 , g2 , . . . , gn }.

(3)

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

3

Table 1 The weight distribution of CD1,2 of Theorem 1. Weight w

Multiplicity Aw

0

1

(q−1)k(3−k) 2

(q−1)2 ((2m−1)k−k2 ) 2

+

(m)

for each 1 ≤ k ≤ m

k

(q − 1)k

Theorem 1. Let symbols and notation be as above. The code CD1,2 defined in (3) is a linear code over Fq with length (m) (q − 1)m + (q − 1)2 2 and dimension m, and its weight distribution is given in Table 1. Proof.

By the definition of D1,2 , it is not hard to see that n = (q − 1)m + (q − 1)2

(m) 2

. For 1 ≤ i ≤ m, let

ei = (ei,1 , ei,2 , . . . , ei,m ) be the vector in Fm q with ei,i = 1 and ei,j = 0 for j ̸ = i. Note that {e1 , e2 , . . . , em } ⊂ D1,2 and G = [g1 g2 . . . gn ]. It follows that Rank(G) = m. Thus the dimension of CD1,2 is equal to m.

For a = (a1 , a2 , . . . , am ) ∈ Fm q , let ca = (a · g1 , a · g2 , . . . , a · gn ) be any codeword in CD1,2 . We now determine the

Hamming weight of ca . Let Na = {x ∈ Fm q : 1 ≤ wt(x) ≤ 2 and a · x = 0}. Then by the orthogonal property of additive characters, we have wt(ca ) = n − Na

=n− =n−

1 ∑ ∑ q

ζpTr(y(a·x))

x∈D1,2 y∈Fq

(4)

1 (∑ ∑ q

y∈Fq

ζ

Tr(y(a·x)) p

+

x∈Fm q

∑ ∑ y∈Fq

wt(x)=1

ζ

Tr(y(a·x)) p

) ,

x∈Fm q wt(x)=2

where Tr denotes the trace function from Fq onto Fp . m Note that for a = (a1 , a2 , . . . , am ) ∈ Fm q and x = (x1 , x2 , . . . , xm ) ∈ Fq , Tr (y(a · x)) =

∑m

i=1

Tr (yai xi ). Assume that

wt(a) = k. It follows that

∑ ∑ y∈Fq

ζpTr(y(a·x)) =

m ∑∑ ∑

Tr(yai xi )

ζp

y∈Fq i=1 xi ∈F∗ q

x∈Fm q wt(x)=1

=

∑ ∑

∑

Tr(yai xi )

ζp

∑ ∑

+

y∈Fq i∈Suppt(a) xi ∈F∗ q

=

∑

y∈Fq

∑∑

∑

1≤i≤m x ∈F∗ i q i∈ / Suppt(a)

1

(5)

χai xi (y) + (m − k)(q − 1)q

i∈Suppt(a) xi ∈F∗ q y∈Fq

= (m − k)(q − 1)q, where χb is an additive character of Fq , b ∈ Fq . Similarly, we have

∑ ∑ y∈Fq

ζpTr(y(a·x)) =

x∈Fm q wt(x)=2

∑ ∑ y∈Fq

=

∑m

ζp

l=1 Tr(yal xl )

x∈Fm q wt(x)=2

∑ ∑ ∑ ∑

(

)

Tr y(ai xi +aj xj )

ζp

∗ y∈Fq 1≤i
=

∑ y∈Fq

+

∑

ζp

(6)

∗ 1≤i
∑ y∈Fq

+

Tr(y(ai xi +aj xj ))

∑ ∑

∑ y∈Fq

∑

∑ ∑

1≤i
∑

∑ ∑

1≤i
(

)

Tr y(ai xi +aj xj )

ζp

1.

4

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

Note that

∑ y∈Fq

∑

∑ ∑

Tr(y(ai xi +aj xj ))

ζp

∗ 1≤i
∑ ∑∑

∑

=

χai xi +aj xj (y)

∗ 1≤i
= 0, and

∑ y∈Fq

∑

∑ ∑

(

)

Tr y(ai xi +aj xj )

ζp

∑

=

1≤i
∑ ∑∑

χai xi +aj xj (y)

1≤i
∑

∑

∑

1≤i
xi ,xj ∈F∗ q

y∈Fq

=

χ0 (y)

ai xi +aj xj =0

( ) k

= q(q − 1)

2

.

It then follows from (6) that

ζpTr(y(a·x)) = q(q − 1)

∑ ∑ y∈Fq

x∈Fm q

( ) k

2

+

∑ y∈Fq

wt(x)=2

( ) = q(q − 1)

k

2

∑ ∑

∑

1

1≤i
+ q(q − 1)2

(

)

m−k 2

(7)

.

Combining (4), (5) and (7), we obtain wt(ca ) = where a ∈

Fm q

(q − 1)k(3 − k)

+

(q − 1)2 ((2m − 1)k − k2 )

,

2 2 with wt(a) = k. The desired weight distribution then follows. This completes the proof.

□

Theorem 2. Let symbols and notation be as above. Then the code CD1,2 defined in (3) is a minimal linear code over Fq . Proof. By the definition of minimal linear codes, in order to prove the minimality of the linear code CD1,2 , we only need to prove that for any nonzero codewords ca = (a · g1 , a · g2 , . . . , a · gn ), cb = (b · g1 , b · g2 , . . . , b · gn ) ∈ CD1,2 , if ca ⪯ cb , then ca and cb are linearly dependent over Fq . m Let a = (a1 , a2 , . . . , am ) ∈ Fm q with wt(a) = s and b = (b1 , b2 , . . . , bm ) ∈ Fq with wt(b) = t. Assume that ca ⪯ cb , i.e. Suppt(ca ) ⊆ Suppt(cb ). Note that D1,2 = {g ∈ Fm q : wt(g) = 1 or wt(g) = 2} = {g1 , g2 , . . . , gn }. For 1 ≤ i ≤ m, let ei = (ei,1 , ei,2 , . . . , ei,m ) be the vector in Fm q with ei,i = 1 and ei,j = 0 for j ̸ = i. For any i ∈ Suppt(a), we have a · ei = ai ̸ = 0. Since ei ∈ D1,2 and ca ⪯ cb , we can deduce that bi = b · ei ̸ = 0. It follows that Suppt(a) ⊆ Suppt(b) and s ≤ t. Now we need to prove that s = t. Suppose on the contrary that s < t. For 1 ≤ i ̸ = j ≤ m, note that λi ei + λj ej ∈ D1,2 , where λi , λj ∈ F∗q . Then for i ∈ Suppt(a), j ∈ Suppt(b)\Suppt(a) and λi , λj ∈ F∗q , we have a · (λi ei + λj ej ) = λi ai + λj aj = λi ai ̸ = 0. It follows from ca ⪯ cb that b · (λi ei + λj ej ) = λi bi + λj bj ̸ = 0 1 holds for all λi , λj ∈ F∗q . Since bj ̸ = 0, if we take λj = −λi bi b− j , then λi bi + λj bj = 0. Thus we arrive at a contradiction. Therefore s = t and Suppt(a) = Suppt(b). For any i, j ∈ Suppt(a) with i ̸ = j, it follows from Suppt(a) = Suppt(b) that bi ̸ = 0 and bj ̸ = 0. Then there exist λi , λj ∈ F∗q such that λi bi + λj bj = 0. Since b · (λi ei + λj ej ) = λi bi + λj bj and ca ⪯ cb , we have that

a · (λi ei + λj ej ) = λi ai + λj aj = 0. It can be deduced that 1 1 aj b − = ai b − j i .

This implies that a = µb for some µ ∈ F∗q . It follows that ca = (a · g1 , a · g2 , . . . , a · gn )

= µcb , where cb = (b · g1 , b · g2 , . . . , b · gn ) and µ ∈ F∗q . That is, ca and cb are linearly dependent over Fq . This completes the proof. □

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

5

The main result of this section is described in the following theorem. Theorem 3. Let m ≥ 3 be a positive integer. Then the code CD1,2 defined in (3) is a [(q − 1)m + (q − 1)2 code over Fq . Furthermore,

(m) , m] minimal linear 2

wmin q−1 ≤ wmax q holds for q > 3. When q = 3,

wmin wmax

2 3

≤

provided that m ≥ 4. When q = 2,

wmin wmax

≤

1 2

provided that m ≥ 6.

Proof. According to Theorem 2, we know that CD1,2 is minimal. For a ∈ Fm q with wt(a) = k, 0 ≤ k ≤ m, it follows from Theorem 1 that wt(ca ) =

(q − 1)k(3 − k)

+

(q − 1)2 ((2m − 1)k − k2 )

2 2 ( )2 (q − 1) (3 + (q − 1)(2m − 1))2 q(q − 1) 3 + (q − 1)(2m − 1) =− k− . + 2 2q 8q

(8)

For simplicity, we denote all nonzero weights of CD1,2 as

wk = −

q(q − 1)

(

2

k−

3 + (q − 1)(2m − 1)

)2

2q

+

(q − 1) (3 + (q − 1)(2m − 1))2 8q

,

(9)

where 1 ≤ k ≤ m. Since m ≥ 3, it can be checked that m 2

<

3 + (q − 1)(2m − 1) 2q

< m.

It then follows from (9) that

wmin = w1 = q − 1 + (q − 1)2 (m − 1). It is easily seen that wmax ≥ w2 = q − 1 + (q − 1)2 (2m − 3). Then we have

wmin w1 q − 1 + (q − 1)2 (m − 1) ≤ = . wmax w2 q − 1 + (q − 1)2 (2m − 3) Note that for q ≥ 3, the inequality q − 1 + (q − 1)2 (m − 1) q − 1 + (q − 1)2 (2m − 3)

≤

(10)

q−1

(11)

q

is equivalent to m≥2+

q q2

− 3q + 2

.

(12)

For q > 3, it is not hard to see that For q = 3, by (10)–(12),

wmin wmax

≤

q−1 q

< 1. It then follows from (10)–(12) that when m ≥ 3, we have provided that m ≥ 4. q q2 −3q+2

wmin wmax

≤

q−1 . q

For q = 2, it follows from (9) that

{ wmax =

w m+1 ,

if m is odd,

w m2 ,

if m is even,

2

(m+1)2

where w m+1 = and w m = 4 2 2 completes the proof. □

m2 +2m . 4

Note that wmin = w1 = m. Thus,

wmin wmax

≤

q−1 q

provided that m ≥ 6. This

Example 1. Let q = 9 and m = 4. Then CD1,2 in Theorem 3 is a minimal linear code over F9 with parameters [416, 4, 200] and weight enumerator 1 + 32z 200 + 384z 328 + 4096z 368 + 2048z 384 , according to Magma [4], which confirms the result of Theorem 3. Obviously, wmin /wmax = 200/384 < 8/9. Example 2. Let q = 5 and m = 6. Then CD1,2 in Theorem 3 is a minimal linear code over F5 with parameters [264, 6, 84] and weight enumerator 1 + 24z 84 + 240z 148 + 1280z 192 + 4096z 204 + 3840z 216 + 6144z 220 , according to Magma [4], which confirms the result of Theorem 3. Obviously, wmin /wmax = 84/220 < 4/5.

6

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840 Table 2 The weight distribution of CD3 of Theorem 4. Weight w

Multiplicity Aw

0 (m) (k) (k) (m−k) (q − 1)3 3 − (q − 1)(q − 2) 3 − (q − 1)2 (m − k) 2 − (q − 1)3 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Remark 1. When q = 2, the result of Theorem 3 is consistent with the result of Theorem 5 given in [15]. 4. The second family of minimal linear codes In this section, let m ≥ 4 be a positive integer. Denote D3 the set of nonzero vectors in Fm q with weight three, i.e. D3 = {g ∈ Fm q : wt(g) = 3} = {g1 , g2 , . . . , gn }. Based on the generic construction given in (1), we obtain a linear code CD3 over Fq . The parameters and the weight distribution of CD3 are given as follows. Lemma 2. Let m ≥ 4 be a positive integer and CD3 be the linear code defined in (1) with the defining set D3 . Then the dimension of CD3 is m. Proof. Let ei = (ei,1 , ei,2 , . . . , ei,m ) be the vector in Fm q with ei,i = 1 and ei,j = 0 for j ̸ = i. Let Span{v1 , v2 , . . . , vn } denote the vector space spanned by the vectors v1 , v2 , . . . , vn in Fm q . When m ≥ 4, for any 1 ≤ i ̸ = j ≤ m, there exists a vector ′ ′ h = (h1 , h2 , . . . , hm ) ∈ Fm q of weight 2 with hi = hj = 0. Let g = ei + h and g = ej + h. It follows that g, g ∈ D3 . Since g + (p − 1)g′ = ei + (p − 1)ej , where p is the characteristic of Fq , then we have Span{ei + (p − 1)ej : 1 ≤ i ̸ = j ≤ m} ⊆ Span{g1 , g2 , . . . , gn }. Note that (e1 + (p − 1)e2 ) + (e1 + (p − 1)e3 ) = e1 + (e1 − e2 − e3 ). Since e1 − e2 − e3 ∈ D3 , then we can deduce that e1 ∈ Span{g1 , g2 , . . . , gn }. Similarly, we can prove that ei ∈ Span{g1 , g2 , . . . , gn } for 2 ≤ i ≤ m. It follows that the dimension of Span{g1 , g2 , . . . , gn } is m. Therefore, the dimension of CD3 is m. This completes the proof. □ Theorem 4. Let symbols (m) and notation be as above. Let m ≥ 4 be a positive integer. Then the code CD3 is a linear code over Fq with length (q − 1)3 3 and dimension m, and its weight distribution is given in Table 2.

(m)

Proof. By the definition of D3 , it is clear that the length of CD3 is 3 (q − 1)3 . According to Lemma 2, we obtain that the dimension of CD3 is m. For a = (a1 , a2 , . . . , am ) ∈ Fm q with wt(a) = k, let ca = (a · g1 , a · g2 , . . . , a · gn ) be any codeword in CD3 , where D3 = {g1 , g2 , . . . , gn }. The weight of ca is equal to n − Na , where Na = {x ∈ Fm q : wt(x) = 3 and a · x = 0}. It follows from the orthogonal property of additive characters of Fq that Na =

=

1 ∑∑ q

ζpTr((a·x)y)

x∈D3 y∈Fq

1∑ ∑ q

y∈Fq

ζpTr((a·x)y)

x∈Fm q wt(x)=3

=

1∑ q

∑

∑ ∑ ∑

Tr((ai xi +aj xj +al xl )y)

ζp

.

∗ ∗ y∈Fq 1≤i
Note that wt(a) = k. When 3 ≤ k ≤ m − 3, we have 0 ≤ |{i, j, l} ∩ Suppt(a)| ≤ 3. It then follows from (13) that Na =

1( q

+

∑

∑ ∑ ∑∑

χai xi +aj xj +al xl (y)

1≤i
∑

∑ ∑ ∑∑

∗ ∗ 1≤i
χai xi +aj xj +al xl (y)

(13)

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

+

∑ ∑ ∑∑

∑

7

χai xi +aj xj +al xl (y)

∗ ∗ 1≤i
+

∑

∑ ∑ ∑∑ ) 1 ,

1≤i
where χai xi +aj xj +al xl is an additive character of Fq . It is not hard to see that the number of solutions (xi , xj , xl ) ∈ F∗q × F∗q × F∗q of the equation ai xi + aj xj + al xl = 0 is equal to (q − 1)2 − (q − 1), where ai , aj , al ∈ F∗q . The number of solutions (xi , xj ) ∈ F∗q × F∗q of the equation ai xi + aj xj = 0 is equal to q − 1, where ai , aj ∈ F∗q . Thus, we have

∑

∑ ∑ ∑∑

( )

χai xi +aj xj +al xl (y) = (q − 1) − (q − 1) q 2

(

)

1≤i
∑

∑ ∑ ∑∑

k

3

,

( ) k (m − k), χai xi +aj xj +al xl (y) = (q − 1)2 q 2

∗ ∗ 1≤i
and

∑

∑ ∑ ∑∑

χai xi +aj xj +al xl (y) = 0.

∗ ∗ 1≤i
It follows that

( ) Na = (q − 1)(q − 2)

k

3

( )

2

+ (q − 1) (m − k)

k

2

+ (q − 1)

3

(

m−k 3

) .

(14)

For k = 1, 2, m − 2, m − 1, m, it can also be verified that Eq. (14) holds. Note that wt(ca ) = n − Na . Hence, wt(ca ) = (q − 1)3 where a ∈

Fm q

( ) m 3

( ) − (q − 1)(q − 2)

k

3

− (q − 1)2 (m − k)

( ) k

2

− (q − 1)3

(

m−k 3

) ,

with wt(a) = k. The desired conclusion then follows. □

Theorem 5. Let q > 2 be a prime power and m ≥ 4 be a positive integer. Then the code CD3 in Theorem 4 is a minimal linear code over Fq . Proof. Similar as the proof of Theorem 2, we need to prove that for any nonzero codewords ca = (a · g1 , a · g2 , . . . , a · gn ), cb = (b · g1 , b · g2 , . . . , b · gn ) ∈ CD3 , if ca ⪯ cb , then ca and cb are linearly dependent over Fq . m Let a = (a1 , a2 , . . . , am ) ∈ Fm q with wt(a) = s and b = (b1 , b2 , . . . , bm ) ∈ Fq with wt(b) = t. Assume that ca ⪯ cb , i.e. Suppt(ca ) ⊆ Suppt(cb ). Note that D3 = {g ∈ Fm : wt(g) = 3 } = { g , 1 g2 , . . . , gn }. For 1 ≤ i ≤ m, let q ei = (ei,1 , ei,2 , . . . , ei,m ) be the vector in Fm q with ei,i = 1 and ei,j = 0 for j ̸ = i. It follows that λi ei + λj ej + λk ek ∈ D3 for pairwise distinct integers i, j, k ∈ {1, 2, . . . , m} and λi , λj , λk ∈ F∗q . We first prove that Suppt(a) ∩ Suppt(b) ̸ = ∅. Suppose on the contrary that Suppt(a) ∩ Suppt(b) = ∅. When s + t ≤ m − 2, for i ∈ Suppt(a), j, k ∈ {1, 2, . . . , m}\(Suppt(a) ∪ Suppt(b)) with j ̸ = k, we have that a · (λi ei +λj ej +λk ek ) = ai λi ̸ = 0. It then follows from ca ⪯ cb that b · (λi ei +λj ej +λk ek ) ̸ = 0, a contradiction. When m − 1 ≤ s + t ≤ m and t ≥ 2, for i, j ∈ Suppt(b), 1 k ∈ Suppt(a) with i ̸ = j and λi , λk ∈ F∗q , there exists λj = −b− j bi λi such that b · (λi ei + λj ej + λk ek ) = bi λi + bj λj = 0. Since ca ⪯ cb , then we have a · (λi ei + λj ej + λk ek ) = ak λk = 0, a contradiction. When m − 1 ≤ s + t ≤ m and t = 1, we can similarly arrive at a contradiction. We then deduce that Suppt(a) ∩ Suppt(b) ̸ = ∅. Secondly, we need to prove that Suppt(a) = Suppt(b) and divide our discussion into the following three cases. Case 1, s ≤ m − 2. For any i ∈ Suppt(a), j, k ̸ ∈ Suppt(a) with j ̸ = k, it follows from λi ei + λj ej + λk ek ∈ D3 that a · (λi ei + λj ej + λk ek ) = λi ai ̸ = 0. Since ca ⪯ cb , then we have b · (λi ei + λj ej + λk ek ) = λi bi + λj bj + λk bk ̸ = 0 holds for all λi , λj , λk ∈ F∗q . That is to say the equation λi bi +λj bj +λk bk = 0 has no solution (λi , λj , λk ) in F∗q × F∗q × F∗q . This implies that |{i, j, k} ∩ Suppt(b)| = 1. Suppose on the contrary that |{i, j, k} ∩ Suppt(b)| ̸ = 1. Then we have the following. Subcase 1.1, i.e. {i, j, k}∩ Suppt(b) = ∅, it is clear that the number of solutions (λi , λj , λk ) ∈ F∗q × F∗q × F∗q of the equation λi bi + λj bj + λk bk = 0 is (q − 1)3 . Subcase 1.2, i.e. |{i, j, k} ∩ Suppt(b)| = 2, without loss of generality we assume that i, j ∈ Suppt(b), then the number of solutions (λi , λj , λk ) ∈ F∗q × F∗q × F∗q of the equation λi bi + λj bj = 0 is (q − 1)2 .

8

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

Subcase 1.3, i.e. |{i, j, k} ∩ Suppt(b)| = 3, it is not hard to see that the number of solutions (λi , λj , λk ) ∈ F∗q × F∗q × F∗q of the equation λi bi + λj bj + λk bk = 0 is (q − 1)2 − (q − 1). Note that q > 2. Thus we arrive at a contradiction. Therefore, |{i, j, k} ∩ Suppt(b)| = 1. Since Suppt(a) ∩ Suppt(b) ̸ = ∅, it can be deduced that i ∈ Suppt(b) and j, k ∈ / Suppt(b). It follows that Suppt(a) ⊆ Suppt(b). Note that for any j ∈ / Suppt(a), we have j ∈ / Suppt(b). That is to say Suppt(b) ⊆ Suppt(a). Therefore, we obtain that Suppt(a) = Suppt(b). Case 2, s = m − 1. Assume that k ∈ / Suppt(a). Then we have k ∈ / Suppt(b). Suppose on the contrary that k ∈ Suppt(b). Note that Suppt(a) ∩ Suppt(b) ̸ = ∅. When t ≤ m − 1, for i ∈ Suppt(a) \ Suppt(b), j ∈ Suppt(a) ∩ Suppt(b) and 1 λi , λj ∈ F∗q , there exists λk = −b− k bj λj such that b · (λi ei + λj ej + λk ek ) = bj λj + bk λk = 0. It follows from ca ⪯ cb that a · (λi ei + λj ej + λk ek ) = ai λi + aj λj = 0 for all λi , λj ∈ F∗q , a contradiction. When t = m, we can similarly arrive at a contradiction. Therefore, we obtain that k ∈ / Suppt(b) and Suppt(b) ⊆ Suppt(a). Suppose that Suppt(b) ̸ = Suppt(a). When t ≥ 2, for i, j ∈ Suppt(b) and l ∈ Suppt(a) \ Suppt(b), we have that b · (λi ei + λj ej + λl el ) = bi λi + bj λj and a · (λi ei + λj ej + λl el ) = ai λi + aj λj + al λl . Note that the number of solutions (λi , λj , λl ) ∈ F∗q × F∗q × F∗q of the equation bi λi + bj λj = 0 is (q − 1)2 and the number of solutions of the equation ai λi + aj λj + al λl = 0 is (q − 1)2 − (q − 1). Since ca ⪯ cb , then we have (q − 1)2 ≤ (q − 1)2 − (q − 1), a contradiction. When t = 1, we can similarly arrive at a contradiction. Therefore, Suppt(a) = Suppt(b). Case 3, s = m. Similarly, we can prove that Suppt(a) = Suppt(b). Now, we prove that ca and cb are linearly dependent over Fq . It follows from Suppt(a) = Suppt(b) that s = t. When s = 1 or 2, it is not hard to see that a = µb for some µ ∈ F∗q . It follows that ca = µcb , where ca = (a · g1 , a · g2 , . . . , a · gn ) and cb = (b · g1 , b · g2 , . . . , b · gn ). When 3 ≤ s ≤ m, for any pairwise distinct integers i, j, k ∈ Suppt(a), it follows from Suppt(a) = Suppt(b) that i, j, k ∈ Suppt(b). Note that λi ei + λj ej + λk ek ∈ D3 , where λi , λj , λk ∈ F∗q . Since ca ⪯ cb , then the solutions (λi , λj , λk ) ∈ F∗q × F∗q × F∗q of the equation b · (λi ei + λj ej + λk ek ) = bi λi + bj λj + bk λk = 0 also satisfy the equation 1 1 1 = b j a− = b k a− a · (λi ei + λj ej + λk ek ) = ai λi + aj λj + ak λk = 0. It can be deduced that bi a− i j k . Suppose on the contrary 1 −1 1 that (bi , bj , bk ) ̸ = µ(ai , aj , ak ) for all µ ∈ F∗q . If bi a− and bk a− are pairwise distinct, then the number of common i , b j aj k solutions (λi , λj , λk ) in F∗q × F∗q × F∗q of the equations

{

bi λi + bj λj + bk λk = 0

(15)

ai λi + aj λj + ak λk = 0

is equal to q − 1. Otherwise, (15) has no solutions in F∗q × F∗q × F∗q . This contradicts the fact that the number of solutions (λi , λj , λk ) ∈ F∗q × F∗q × F∗q of bi λi + bj λj + bk λk = 0 is (q − 1)2 − (q − 1). Hence, (bi , bj , bk ) = µ(ai , aj , ak ) for some µ ∈ F∗q . It follows that b = µa and cb = µca . Thus, ca and cb are linearly dependent over Fq . This completes the proof. □ The main result of this section is described in the following theorem. Theorem 6. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD3 in Theorem 4 is a [(q − 1)3 minimal linear code over Fq . Furthermore,

(m) , m] 3

wmin q−1 ≤ wmax q holds for q ≥ 7. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4 or 5,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Proof. By Theorem 5, we know that CD3 is minimal over Fq . For a ∈ Fm q with wt(a) = k, 1 ≤ k ≤ m, it follows from Theorem 4 that

( ) m

wt(ca ) = (q − 1)3

3

( ) ( ) ( ) k k m−k − (q − 1)(q − 2) − (q − 1)2 (m − k) − (q − 1)3 . 3

2

3

For simplicity, we denote all nonzero weights of CD3 as

wk = (q − 1)3

( ) m 3

( ) − (q − 1)(q − 2)

where 1 ≤ k ≤ m. Note that w1 = (q −

k

3

m 1)3 3

( )

( ) ( ) k m−k − (q − 1)2 (m − k) − (q − 1)3 , 2

− (q −

m−1 1)3 3

(

)

wmin w1 3(q − 1)2 ) . ≤ = ( wmax wm (q − 1)2 − (q − 2) m Note that the inequality m≥

3q(q − 1) q2 − 3q + 3

.

3(q−1)2

((q−1)2 −(q−2))m

≤

q−1 q

and wm = (q −

3

m 1)3 3

( )

− (q − 1)(q − 2)

(m) 3

. Then we have (16)

is equivalent to (17)

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

9

It can be deduced that 3q(q − 1) ≤4 q2 − 3q + 3 holds for q ≥ 8. It then follows from (16) and (17) that when q ≥ 8, q−1 wmin ≤ wmax q holds for m ≥ 4. According to (16), (17) and the weight distribution of CD3 given in Theorem 4, we obtain that when w w w q−1 q−1 q−1 q = 7, w min ≤ q holds for m ≥ 4. When q = 4 or 5, w min ≤ q holds for m ≥ 5. Moreover, when q = 3, w min ≤ q max max max holds for m ≥ 6. This completes the proof. □ Example 3. Let q = 3 and m = 7. Then the code CD3 in Theorem 6 is a minimal linear code over F3 with parameters [280, 7, 120] and weight enumerator 1 + 14z 120 + 1204z 180 + 280z 198 + 560z 192 + 128z 210 according to Magma [4], which confirms the result of Theorem 6. Obviously, wmin /wmax = 120/210 < 2/3. Example 4. Let q = 4 and m = 6. Then the code CD3 in Theorem 6 is a minimal linear code over F4 with parameters [540, 6, 270] and weight enumerator 1 + 18z 270 + 1458z 390 + 135z 396 + 1215z 408 + 729z 420 + 540z 426 according to Magma [4], which confirms the result of Theorem 6. Obviously, wmin /wmax = 270/426 < 3/4. 5. More families of minimal linear codes In this section, based on the minimal linear codes constructed in Sections 3 and 4, we obtain more families of minimal linear codes over Fq with wmin /wmax ≤ (q − 1)/q. The following lemma plays an important role in this section.

˜ Lemma 3. Let D and ˜ D be two subsets of Fm D be two linear codes q . Assume that D = {g1 , g2 , . . . , gn } and D ⊆ D. Let CD and C˜ ˜ over Fq defined in (1) with the defining sets D and D, respectively. If CD is a minimal linear code over Fq with dimension m, then C˜ D is also minimal. Proof. Let ˜ n be a positive integer with ˜ n ≥ n. Suppose that

˜ D = {g1 , g2 , . . . , gn , gn+1 , . . . , g˜ n }. ca ⪯ ˜ cb , then we have For any nonzero codewords ˜ ca = (a · g1 , a · g2 , . . . , a · g˜ cb = (b · g1 , b · g2 , . . . , b · g˜ n ), ˜ n ) ∈ C˜ D , if ˜ Suppt(˜ ca ) ⊆ Suppt(˜ cb ). It follows that

{1, 2, . . . , n} ∩ Suppt(˜ ca ) ⊆ {1, 2, . . . , n} ∩ Suppt(˜ cb ).

(18)

Note that D = {g1 , g2 , . . . , gn }. We can deduce that ca = (a · g1 , a · g2 , . . . , a · gn ), cb = (b · g1 , b · g2 , . . . , b · gn ) ∈ CD . It is clear that Suppt(ca ) = {1, 2, . . . , n} ∩ Suppt(˜ ca ) and Suppt(cb ) = {1, 2, . . . , n} ∩ Suppt(˜ cb ). Then by (18), we have Suppt(ca ) ⊆ Suppt(cb ). Since CD is a minimal linear code over Fq , then we deduce that ca and cb are linearly dependent over Fq . That is, ca = µcb for some µ ∈ F∗q . It follows that ca − µcb = 0. Note that ca − µcb = ((a − µb) · g1 , (a − µb) · g2 , . . . , (a − µb) · gn ) and the dimension of CD is m. Then we have a − µb = 0, i.e. a = µb. Thus,

˜ ca = (a · g1 , a · g2 , . . . , a · gn , . . . , a · g˜ n) = µ(b · g1 , b · g2 , . . . , b · gn , . . . , b · g˜n ) = µ˜ cb . It follows that ˜ ca and ˜ cb are linearly dependent over Fq . By the definition of minimal linear code, C˜ D is minimal over Fq . This completes the proof. □ Denote D1,3 = {g ∈ Fm q : wt(g) = 1 or wt(g) = 3}, D2,3 = {g ∈ Fm q : wt(g) = 2 or wt(g) = 3}, and D1,2,3 = {g ∈ Fm q : 1 ≤ wt(g) ≤ 3}. Let CD1,3 , CD2,3 and CD1,2,3 be the linear codes defined in (1) with the defining sets D1,3 , D2,3 and D1,2,3 , respectively. Then we have the following results.

10

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840 Table 3 The weight distribution of CD1,3 of Theorem 7. Weight w

Multiplicity Aw

0 (m−k) (m) (k) (k) k(q − 1) + (q − 1)3 3 − (q − 1)(q − 2) 3 − (q − 1)2 (m − k) 2 − (q − 1)3 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Table 4 The weight distribution of CD2,3 of Theorem 8. Weight w

Multiplicity Aw

0 ((m) (m−k) (k)) ((m) (m−k)) (q − 1)2 2 − 2 − (m − k) 2 + (q − 1)3 3 − 3 (k) (k) −(q − 1) 2 − (q − 1)(q − 2) 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Theorem 7. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD1,3 is a [(q − 1)m + (q − 1)3 minimal linear code over Fq , and its weight distribution is given in Table 3. Furthermore,

(m) 3

, m]

q−1 wmin ≤ wmax q holds for q ≥ 7. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4 or 5,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Proof. Note that D3 ⊆ D1,3 and CD3 is a minimal linear code over Fq with dimension m. It follows from Lemma 3 that C(D1),3 m is minimal and the dimension of CD1,3 is m. By the definition of D1,3 , it is clear that the length of CD1,3 is (q−1)m+(q−1)3 3 . m Let ca = (a · g1 , a · g2 , . . . , a · gn ) be any codeword in CD1,3 , where a ∈ Fq with wt(a) = k and D1,3 = {g1 , g2 , . . . , gn }. From the proof of Theorems 1 and 4, we can obtain the weight of ca as follows: wt(ca ) = n −

=n−

1 ∑ ∑ q

ζpTr((a·x)y)

x∈D1,3 y∈Fq

1( ∑ ∑ q

ζpTr((a·x)y) +

∑ ∑

ζpTr((a·x)y)

)

y∈Fq x∈Fm q wt(x)=3

y∈Fq x∈Fm q wt(x)=1

( ) ( ) m k = k(q − 1) + (q − 1)3 − (q − 1)(q − 2) 3 3 ( ) ( ) k m−k − (q − 1)2 (m − k) − (q − 1)3 . 2

3

For simplicity, we denote all nonzero weights of CD1,3 as

wk = k(q − 1) + (q − 1)

3

( ) m 3

( ) − (q − 1)(q − 2)

k

3

2

( )

− (q − 1) (m − k)

k

2

− (q − 1)

3

(

m−k 3

) ,

where 1 ≤ k ≤ m. It is not hard to see that

w1 6 + 3(q − 1)2 (m − 1)(m − 2) = wm 6m + ((q − 1)2 − (q − 2))m(m − 1)(m − 2) 3(q − 1)2 . ≤ ((q − 1)2 − (q − 2))m From the proof of Theorem 6, we obtain that

≤

q−1 q

holds for m ≥ 4, q ≥ 8. It follows from

wmin wmax q−1

≤

w1 wm

min ≤ q−q 1 holds for m ≥ 4, q ≥ 8. Moreover, with the help of the weight distribution of CD1,3 , wwmax ≤ q holds wmin wmin q−1 q−1 for m ≥ 4, q = 7, and when q = 4 or 5, w ≤ holds for m ≥ 5. When q = 3, ≤ holds for m ≥ 6. This q wmax q max

that

wmin wmax

3(q−1)2 ((q−1)2 −(q−2))m

completes the proof. □

Theorem 8. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD2,3 is a [(q − 1)2 minimal linear code over Fq , and its weight distribution is given in Table 4. Furthermore,

(m) 2

+ (q − 1)3

wmin q−1 ≤ wmax q holds for q ≥ 5. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

(m) 3

, m]

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

11

Table 5 The weight distribution of CD1,2,3 of Theorem 9. Weight w

Multiplicity Aw

0 ((m) (m−k)) ((m) (m−k) (k)) (q − 1)2 2 − 2 − (m − k) 2 + (q − 1)3 3 − 3 ( (k) (k)) +(q − 1) k − 2 − (q − 2) 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Proof. The proof of this theorem is similar to that of Theorem 7. Note that D3 ⊆ D2,3 and CD3 is a minimal linear code over Fq with dimension m. According to Lemma 3, CD2,3 is minimal and the dimension of CD2,3 is m. From the proof of Theorems 1 and 4, we can obtain the weight distribution of CD2,3 similarly. The desired conclusion on the ratio of wmin to wmax then follows from the proof of Theorem 6 and the weight distribution of CD2,3 . We omit the details of the proof. □ Similar as the proof of Theorems 7 and 8, we obtain the following result. 2 Theorem (m9. ) Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD1,2,3 is a [(q − 1)m + (q − 1) (q − 1)3 3 , m] minimal linear code over Fq , and its weight distribution is given in Table 5. Furthermore,

(m) 2

+

q−1 wmin ≤ wmax q holds for q ≥ 5. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Example 5. Let q = 3 and m = 6. Then the code CD1,3 in Theorem 7 is a minimal linear code over F3 with parameters [172, 6, 82] and weight enumerator 1 + 12z 82 + 60z 116 + 160z 120 + 240z 112 + 192z 110 + 64z 132 according to Magma [4], which confirms the result of Theorem 7. Obviously, wmin /wmax = 82/132 < 2/3. Example 6. Let q = 5 and m = 4. Then the code CD2,3 in Theorem 8 is a minimal linear code over F5 with parameters [352, 4, 240] and weight enumerator 1 + 16z 240 + 512z 280 + 96z 300 according to Magma [4], which confirms the result of Theorem 8. Obviously, wmin /wmax = 240/300 = 4/5. Example 7. Let q = 3 and m = 6. Then the code CD1,2,3 in Theorem 9 is a minimal linear code over F3 with parameters [232, 6, 102] and weight enumerator 1 + 12z 102 + 252z 150 + 240z 156 + 224z 162 according to Magma [4], which confirms the result of Theorem 9. Obviously, wmin /wmax = 102/162 < 2/3. Now, we consider the puncture version of CD . We have the following theorem. Theorem 10. Let D = {g1 , g2 , . . . , gn } ⊆ Fm q and D = {g1 , g2 , . . . , gn¯ } be a subset of D such that D. That is

⋃

λ∈F∗q

λD is a partition of

D = {λg : λ ∈ F∗q , g ∈ D}, where for each pair of distinct elements gi , gj ∈ D, 1 ≤ i ̸ = j ≤ n, we have gi ̸ = λgj for all λ ∈ F∗q . Let CD and CD be two linear codes over Fq defined in (1) with the defining sets D and D, respectively. If CD is a minimal linear code over Fq , then CD is also minimal. Proof. For any nonzero codewords c¯ a = (a · g1 , a · g2 , . . . , a · gn¯ ), c¯ b = (b · g1 , b · g2 , . . . , b · gn¯ ) ∈ CD , if c¯ a ⪯ c¯ b , then we have Suppt(c¯ a ) ⊆ Suppt(c¯ b ). Now we need to prove that Suppt(ca ) ⊆ Suppt(cb ). Suppose on the contrary ⋃ that Suppt(ca ) ̸ ⊆ Suppt(cb ). There exists gl ∈ D such that a · gl ̸ = 0 and b · gl = 0, where n¯ + 1 ≤ l ≤ n. Since D = λ∈F∗ λD, q

1 then there exists a unique λl ∈ F∗q such that gl ∈ λl D. It follows that gl = λl gi , where gi ∈ D. Note that a · gi = λ− l a · gl ̸ = 0 −1 and b · gi = λl b · gl = 0. This contradicts the fact that Suppt(c¯ a ) ⊆ Suppt(c¯ b ). Therefore, we have Suppt(ca ) ⊆ Suppt(cb ), i.e. ca ⪯ cb . Since CD is a minimal linear code over Fq . Then it can be deduced that ca = µcb for some µ ∈ F∗q . Note that ca = (a · g1 , a · g2 , . . . , a · gn ) and cb = (b · g1 , b · g2 , . . . , b · gn ). Then we have c¯ a = µ¯cb . By the definition of minimal linear code, CD is minimal over Fq . This completes the proof. □

12

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840 Table 6 The weight distribution of CD1,2 of Corollary 1. Weight w

Multiplicity Aw

0

1

k(3−k) 2

+

(q−1)((2m−1)k−k2 ) 2

(m)

for each 1 ≤ k ≤ m

k

(q − 1)k

Table 7 The weight distribution of CD3 of Corollary 2. Weight w

Multiplicity Aw

0 (m) (k) (k) (m−k) (q − 1)2 3 − (q − 2) 3 − (q − 1)(m − k) 2 − (q − 1)2 3 for each 1 ≤ k ≤ m

(1m) k

(q − 1)k

Table 8 The weight distribution of CD1,3 of Corollary 3. Weight w

Multiplicity Aw

0 (m) (k) (k) (m−k) k + (q − 1)2 3 − (q − 2) 3 − (q − 1)(m − k) 2 − (q − 1)2 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Recall that D1,2 = {g ∈ Fm = 2}. It is not hard to see that for any λ ∈ F∗q , wt(g) = wt(λg). Then q : wt(g) = 1 or wt(g) ⋃ we can choose a subset D1,2 of D1,2 such that λ∈F∗ λD1,2 is just a partition of D1,2 . That is D1,2 = {λg : λ ∈ F∗q , g ∈ D1,2 }, q

where for each pair of distinct elements g1 , g2 ∈ D1,2 , we have g1 ̸ = λg2 for all λ ∈ F∗q . Thus, the linear code CD1,2 can be punctured into a shorter linear code CD1,2 with the defining set D1,2 . Similarly, we can choose the subsets D3 , D1,3 , D2,3 and D1,2,3 such that

⋃

D3 =

λD3 , D1,3 =

λ∈F∗q

and D1,2,3 =

⋃

λ∈F∗q

⋃

λD1,3 , D2,3 =

λ∈F∗q

⋃

λD2,3 ,

λ∈F∗q

λD1,2,3 . Let CD3 , CD1,3 , CD2,3 and CD1,2,3 be the linear codes over Fq defined in (1) with the defining sets

D3 , D1,3 , D2,3 and D1,2,3 , respectively. Then the linear codes CD3 , CD1,3 , CD2,3 and CD1,2,3 can be punctured into shorter linear codes CD3 , CD1,3 , CD2,3 and CD1,2,3 , respectively. According to Theorems 3 and 6–10, we can obtain the following corollaries directly. Corollary 1. Let m ≥ 3 be a positive integer. Then the code CD1,2 is a [m + (q − 1) its weight distribution is given in Table 6. Furthermore,

(m) 2

, m] minimal linear code over Fq and

wmin q−1 ≤ wmax q holds for q > 3. When q = 3,

wmin wmax

≤

2 3

provided that m ≥ 4. When q = 2,

wmin wmax

≤

1 2

provided that m ≥ 6.

Corollary 2. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD3 is a [(q − 1)2 linear code over Fq , and its weight distribution is given in Table 7. Furthermore,

(m) , m] minimal 3

wmin q−1 ≤ wmax q holds for q ≥ 7. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4 or 5,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Corollary 3. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD1,3 is a [m + (q − 1)2 minimal linear code over Fq , and its weight distribution is given in Table 8. Furthermore,

wmin q−1 ≤ wmax q holds for q ≥ 7. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4 or 5,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

(m) 3

, m]

Z. Shi and F.-W. Fu / Discrete Mathematics 343 (2020) 111840

13

Table 9 The weight distribution of CD2,3 of Corollary 4. Weight w

Multiplicity Aw

0 ((m) (m−k) (k)) ((m) (m−k)) (q − 1) 2 − 2 − (m − k) 2 + (q − 1)2 3 − 3 (k) (k) − 2 − (q − 2) 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Table 10 The weight distribution of CD1,2,3 of Corollary 5. Weight w

Multiplicity Aw

0 ((m) (m−k)) ((m) (m−k) (k)) (q − 1) 2 − 2 − (m − k) 2 + (q − 1)2 3 − 3 (k) (k) +k − 2 − (q − 2) 3 for each 1 ≤ k ≤ m

1

(m) k

(q − 1)k

Corollary 4. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD2,3 is a [(q − 1) minimal linear code over Fq , and its weight distribution is given in Table 9. Furthermore,

(m) 2

+ (q − 1)2

(m) 3

, m]

wmin q−1 ≤ wmax q holds for q ≥ 5. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Corollary 5. Let m ≥ 4 be a positive integer and q > 2 be a prime power. Then the code CD1,2,3 is a [m + (q − 1)

(m) 1)2 3 , m] minimal linear code over Fq , and its weight distribution is given in Table 10. Furthermore,

(m) 2

+ (q −

wmin q−1 ≤ wmax q holds for q ≥ 5. When q = 3,

wmin wmax

≤

q−1 q

provided that m ≥ 6. When q = 4,

wmin wmax

≤

q−1 q

provided that m ≥ 5.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are very grateful to the reviewers and the Editor for their comments that improved the quality and presentation of this paper. This research is supported by the National Natural Science Foundation of China under Grant Nos. 61571243 and 61971243. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

A. Ashikhmin, A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory 44 (5) (1998) 2010–2017. D. Bartoli, M. Bonini, Minimal linear codes in odd characteristic, IEEE Trans. Inform. Theory (2019) http://dx.doi.org/10.1109/TIT.2019.2891992. M. Bonini, M. Borello, Minimal linear codes arising from blocking sets, 2019, ArXiv preprint, arXiv:1907.04626. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language. Computational algebra and number theory, J. Symbolic Comput. 24 (3–4) (1997) 235–265. C. Carlet, C. Ding, J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory 51 (6) (2005) 2089–2102. H. Chabanne, G. Cohen, A. Patey, Towards secure two-party computation from the wire-tap channel, in: International Conference on Information Security and Cryptology-ICISC 2013, in: LNCS, vol. 8565, 2014, pp. 34–46. S. Chang, J.Y. Hyun, Linear codes from simplicial complexes, Des. Codes Cryptogr. 86 (10) (2017) 2167–2181. C. Ding, Linear codes from some 2-designs, IEEE Trans. Inform. Theory 61 (6) (2015) 3265–3275. C. Ding, Z. Heng, Z. Zhou, Minimal binary linear codes, IEEE Trans. Inform. Theory 64 (10) (2018) 6536–6545. Z. Heng, C. Ding, Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl. 54 (2018) 176–196. R. Lidl, H. Niederreiter, Finite Fields, Cambridge University Press, 1997. J.L. Massey, Minimal codewords and secret sharing, in: Proceedings of the 6th Joint Swedish-Russian Workshop on Information Theory, 1993, pp. 246–249. S. Mesnager, A. Sınak, Several classes of minimal linear codes with few weights from weakly regular plateaued functions, 2018, arXiv:1808.03877. J. Yuan, C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inform. Theory 52 (1) (2006) 206–212. W. Zhang, H. Yan, H. Wei, Four families of minimal binary linear codes with wmin /wmax ≤ 1/2, Appl. Alg. Eng. Commun. Comput. 30 (2) (2019) 175–184. Z. Zhou, C. Tang, X. Li, C. Ding, Binary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inform. Theory 65 (1) (2019) 16–27.