# Minimal linear codes from characteristic functions

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of F_q to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of F_q, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, DOI: 10.1109/TIT.2019.2918537]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method.

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## I Introduction

Throughout this paper, let be a prime and , where is a positive integer. Let be the finite field with elements and let be the multiplicative group of . An linear code over is a -dimension subspace of with minimum (Hamming) distance . Let be the number of codewords with Hamming weight in . The weight enumerator of is the polynomial and the weight distribution of is . The minimum distance determines the error-correcting capability of

. The weight distribution contains important information for estimating the probability of error detection and correction. Hence, the weight distribution attracts much attention in coding theory and much work focus on the determination of the weight distributions of linear codes. Let

be the number of nonzero in the weight distribution. Then the code is called a -weight code. Linear codes can be applied in consumer electronics, communication and data storage system. Linear codes with few weights are important in secret sharing [9, 34], authentication codes [21, 24], association schemes [3] and strongly regular graphs [4].

For a vector

, let be the support of and let be the Hamming weight of . Note that . A vector covers a vector if . A codeword in a linear code is minimal if covers only the codeword for all , but no other codewords in . A linear code is minimal if any codeword of is minimal. Minimal linear codes have interesting applications in secret sharing schemes [9, 23, 28, 34] and secure two-party computation [2, 14, 17]. A sufficient condition for a linear code to be minimal is given in the following lemma.

###### Lemma I.1

[1] A linear code over is minimal if , where and denote the minimum and maximum nonzero Hamming weights in the code respectively.

Some minimal linear codes with few weights can be constructed by the defining set method [19, 20]. Let be a subset of . Then a linear code of length defined over is defined by

 CD={(Tr(βx))x∈D:β∈Fq}, (1)

where is called the defining set of and is the trace function from to . From this construction, many linear codes can be constructed by different choices of . These minimal linear codes from this method satisfy the sufficient condition . This sufficient condition is not necessary [7]. Chang and Hyun [15] made a breakthrough and constructed an infinite family of minimal binary linear codes with . Heng et al. [25] presented a sufficient and necessary condition for minimal linear codes in the following theorem.

###### Theorem I.2

Let be a linear code over . Then is minimal if and only if

 ∑c∈F∗pwt(a+cb)≠(p−1)wt(a)−wt(b) (2)

for any linearly independent codewords .

They also constructed an infinite family of minimal ternary linear codes with . Ding et al. [22] presented more necessary and sufficient conditions for minimal binary linear codes and constructed three infinite families of minimal binary linear codes. Zhang et al. [35] constructed four families of minimal binary linear codes from Krawtchouk polynomials. Xu and Qu [33]

studied minimal linear codes for odd

and presented three infinite families of minimal linear codes. These minimal linear codes are constructed from the following method [8, 15, 30, 32]. Let be a function from to such that

 {f(0)=0,f(x)≠Tr(wx) for all w∈Fq. (3)

A linear code over can be defined by

 Cf={(uf(x)−Tr(vx))x∈F∗q:u∈Fp,v∈Fq} (4)

By the choice of , many linear codes with good properties can be defined.

Inspired by these recent results, we use the characteristic function of a subset of in constructing minimal linear codes in (4). For binary case, by a simple property of characteristic functions, we can present more minimal binary linear codes from known minimal binary linear codes. Furthermore, we employ characteristic functions corresponding to some subspaces to construct minimal linear codes, which generalize [22] and [33]. The rest of this paper is organized as follows. In Section 2, we present some basic results on -ary functions, Krawchouk polynomials, and minimal linear codes. In Section 3, we present more minimal linear codes from characteristic functions. In section 4, we use characteristic functions to present a characterization of minimal linear codes from the defining set method. Section 5 makes a conclusion.

## Ii Preliminaries

In this section, we will introduce some results on -ary functions, Krawchouk polynomials, and minimal linear codes.

### Ii-a p-ary functions

A -ary function is a function from or to . The Walsh transform of a -ary function at a point is defined by

 ^f(w):=∑x∈Fqζf(x)−Tr(wx)p,

where is the primitive -th root of unity and is the trace function from to . The Walsh transform of a -ary function at a point is defined by

 ^f(w):=∑x∈Fmpζf(x)−⟨w,x⟩p,

where is the inner product of . A function is called a -ary bent function, if for any . When , a -ary (bent) function is just a Boolean (bent) function.

An important class of Boolean function is the general Maiorana-McFarland class, which can be used to generate Boolean functions with good cryptographic properties [7, 10, 18, 27, 29]. Let be a positive integer and let be two positive integers such that . The function in the general Maiorana-McFarland class has the form

 f(x,y)=ϕ(x)⋅y+g(x), (5)

where , , is a mapping from to , and is a Boolean function in variables.

Krawchouk polynomials [13, 26] are useful in bent functions and coding theory. Let be a positive integer. The Krawchouk polynomial is defined by

 Pk(x)=k∑j=0(−1)j(xj)(m−xk−j), (6)

where . The Krawchouk polynomials satisfy

• ,

• ,

• ,

• for ,

• ,

• .

• , where such that .

### Ii-B Linear codes

In this subsection, we present some results on linear codes defined in (4).

Parameters of binary linear codes in (4) can be determined by the following Theorem.

###### Theorem II.1 ([22])

Let and let be defined in (4) from a Boolean function satisfying (3). The code has length and dimension . The weight distribution of is given by the following multiset union

 {q−^f(w)2:w∈Fq}∪{2m−1:w∈F∗q}∪{0}.

A necessary and sufficient condition of a minimal binary linear code in (4) is given in the following theorem, which is more efficient than Theorem I.2.

###### Theorem II.2 ([22])

Let and let be defined in (4) from a Boolean function satisfying (3). Then is minimal if and only if and for every pair of distinct vectors and in .

Let be the -th cyclotomic field over the rational field . Then the field extension is Galois of degree and the Galois group is , where is an automorphism of defined by . Parameters of a linear code in (4) for odd can be given in the following lemma.

###### Lemma II.3 ([30])

Let be an odd prime and let be defined in (4). Then is a code and the Hamming weight of a codeword is given by

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0,if u=0,v=0;pm−pm−1,if u=0,v≠0;pm−pm−1−1p∑w∈F∗pσw(σu(^f(u−1v))),otherwise.

## Iii Minimal linear codes from characteristic functions

In this section, we will present some minimal linear codes from characteristic functions associated with different subsets of .

Let . The characteristic function of is

 fD(x)={1,if x∈D0,otherwise.

By the characteristic function , a linear code can be constructed by

 CfD={(ufD(x)−Tr(vx))x∈F∗q:u∈Fp,v∈Fq}. (7)

We first give some properties of characteristic functions.

###### Lemma III.1

Let and let . Then

 ^fD(w)+^f¯¯¯¯D(w)={(q−1)ζp+q+1,if w=0,1−ζp,otherwise.
###### Proof:
 ^fD(w)= ∑x∈FqζfD(x)−Tr(wx)p = ∑x∈DζfD(x)−Tr(wx)p+∑x∈Fq∖Dζ−Tr(wx)p = ∑x∈D(ζfD(x)−Tr(wx)p−ζ−Tr(wx)p)+∑x∈Fqζ−Tr(wx)p = (ζp−1)∑x∈Dζ−Tr(wx)p+∑x∈Fqζ−Tr(wx)p

and

 ^f¯¯¯¯D(w)=(ζp−1)∑x∈¯¯¯¯Dζ−Tr(wx)p+∑x∈Fqζ−Tr(wx)p.

Then

 ^fD(w)+^f¯¯¯¯D(w)= (ζp−1)(∑x∈Dζ−Tr(wx)p+∑x∈¯¯¯¯Dζ−Tr(wx)p)+2∑x∈Fqζ−Tr(wx)p = (ζp−1)(∑x∈Fqζ−Tr(wx)p−1)+2∑x∈Fqζ−Tr(wx)p = 1−ζp+(ζp+1)∑x∈Fqζ−Tr(wx)p.

Since for any , this lemma follows.

When , we have for any . If , , we also have for any . By Theorem II.1 and Lemma III.1, we have the following corollary.

###### Corollary III.2

Let . Let and such that their characteristic functions and satisfy (3). Then the code has length and dimension . The weight distribution of is given by the following multiset union

 {q−^f¯¯¯¯D(w)2:w∈Fq}∪{2m−1:w∈F∗q}∪{0} = {q−2+^fD(w)2:w∈Fq}∪{2m−1:w∈F∗q}∪{0}.
###### Remark 1

Let be bent or semi-bent function satisfying (3). Let . The Walsh transforms of are given in [16]. By Theorem II.2. Hence, the codes and are minimal. They satisfy that .

By Lemma II.3 and Lemma III.1, we have the following corollary.

###### Corollary III.3

Let be an odd prime. Let and such that their characteristic functions and satisfy (3). Then is a code and the Hamming weight of a codeword is given by

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩0,if u=0,v=0;pm−pm−1,if u=0,v≠0;pm−pm−1−1p∑w∈F∗pσw(σu(^f¯¯¯¯D(0))),if u≠0,v=0;pm−pm−1−1p∑w∈F∗pσw(σu(^f¯¯¯¯D(u−1v))),if u≠0,v≠0. = ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩0,if u=0,v=0;pm−pm−1,if u=0,v≠0;pm−1−1+1p∑w∈F∗pσw(σu(^fD(0))),if u≠0,v=0;pm−pm−1−1+1p∑w∈F∗pσw(σu(^fD(u−1v))),if u≠0,v≠0.
###### Remark 2

By Corollary III.2 and Corollary III.3, we can obtain from known .

###### Lemma III.4

Let and such that . Let . Then

 ^fD(w)={^fD1(w)+^fD2(w)−q,if w=0,^fD1(w)+^fD2(w),otherwise.
###### Proof:

Note that

 ^fD1(w)= (ζp−1)∑x∈D1ζ−Tr(wx)p+∑x∈Fqζ−Tr(wx)p, ^fD2(w)= (ζp−1)∑x∈D2ζ−Tr(wx)p+∑x∈Fqζ−Tr(wx)p, ^fD(w)= (ζp−1)∑x∈Dζ−Tr(wx)p+∑x∈Fqζ−Tr(wx)p.

By , we have this lemma.

### Iii-a Some minimal binary linear codes from known minimal binary linear codes

In this subsection, we obtain more minimal binary linear codes from known minimal binary linear codes.

We will present more minimal binary linear codes from minimal binary linear codes in [22].

###### Theorem III.5

Let be an odd integer, , and . Let and let . Let be the Boolean function defined in (5), where , is an injection from to , and for any . Let . The code defined in (7) is a minimal code with . The weight distribution of is given in Table I (resp. Table II) when is odd (resp. even).

###### Proof:

Note that

where runs from to , and . By Lemma III.1, we have

Note that for any pair of distinct , . By Theorem II.2, The code is minimal. By Theorem II.1, we have the weight distribution of . Note that and . Then . This theorem follows.

###### Theorem III.6

Let be a positive integer and let . The code defined in (7) has length , dimension , and the weight distribution in Table III.

###### Proof:

Note that

where are Krawchouk polynomials defined in (6). By Theorem II.1, we have the distribution of in Table III.

###### Remark 3

By Theorem II.2, conditions of to be minimal can be obtained.

### Iii-B Minimal linear codes from characteristic functions corresponding to subspaces

In this subsection, we will give some minimal linear codes from characteristic functions corresponding to some subspaces.

Let be subspaces of such that

 (8)

where . (8) holds if and only if one of the following conditions holds:

(i) ;

(ii) and ;

(iii) is even and .

Let . Note that

 ^fD(w)= ∑x∈FqζfD(x)−Tr(wx)p = ∑x∈DζfD(x)−Tr(wx)p+∑x∈Fq∖Dζ−Tr(wx)p = ∑x∈D(ζfD(x)−Tr(wx)p−ζ−Tr(wx)p)+∑x∈Fqζ−Tr(wx)p = s∑i=1(ζp−1)(∑x∈Eiζ−Tr(wx)p−1)+∑x∈Fqζ−Tr(wx)p = ⎧⎪ ⎪⎨⎪ ⎪⎩pm+(ζp−1)(∑si=1|Ei|−s),if w=0;(ζp−1)(|Ei|−s),if w∈E⊥i∖{0} %for 1≤i≤s;−(ζp−1)s,if w∈Fq∖(∪si=1E⊥i), (9)

where . Then we have linear codes and in the following theorem.

###### Theorem III.7

Let , where satisfy (8). Let and be defined in (7). Then and are codes with the weight distributions in Table IV and Table V, respectively.

###### Proof:

By (III-B), Theorem II.1 and Corollary III.2, for , we have the weight distributions of and . By (III-B), Lemma II.3 and Corollary III.3, for odd, we have the weight distributions of and . Hence, this theorem follows.

By choosing different subspaces in Theorem III.7, we can obtain many minimal codes, in which we can find minimal codes with . When and is even, we have the following theorem.

###### Theorem III.8 (Theorem 18, [22])

Let , be even, and . Let , where satisfy (8), . Then and are minimal if and only if . Furthermore, if , the code satisfies that . If , then satisfies that .

###### Theorem III.9

Let be odd and be even. Let , where satisfy (8) and . If (resp. ), then (resp. ) is minimal. Furthermore, if (resp. ), the code (resp. ) satisfies that .

###### Proof:

By the weight distribution of in Table IV, we have weights of nonzero codewords of : , , , and . Obviously, . Let for . Take two linearly independent codewords , where and . Note that

 a∈H1 if and only if u1≠0,v1=0; a∈H2 if and only if