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Research article

Weight distributions for projective binary linear codes from Weil sums

  • Received: 28 January 2021 Accepted: 01 June 2021 Published: 07 June 2021
  • MSC : 94B15, 11T71

  • A class of projective binary linear codes are constructed and their weight distributions are investigated using Weil sums. They have at most three nonzero weights, containing some optimal codes. Their dual codes are also studied and some of them are either optimal or almost optimal.

    Citation: Shudi Yang, Zheng-An Yao. Weight distributions for projective binary linear codes from Weil sums[J]. AIMS Mathematics, 2021, 6(8): 8600-8610. doi: 10.3934/math.2021499

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  • A class of projective binary linear codes are constructed and their weight distributions are investigated using Weil sums. They have at most three nonzero weights, containing some optimal codes. Their dual codes are also studied and some of them are either optimal or almost optimal.



    Throughout this paper, we let q=2m for a positive integer m. An [n,κ,d] linear code C over the finite field F2 is a κ-dimensional subspace of Fn2 with minimum distance d. A linear code C is called projective if its dual code has minimum distance at least 3. For a codeword cC the Hamming weight wt(c) is the number of nonzero coordinates in c. Let Ai be the number of codewords with weight i in C of length n. The sequence (1,A1,,An) is referred as the weight distribution of C. If the number of nonzero Ai in the sequence (A1,,An) is equal to t, we call C a t-weight code.

    The weight distribution contains important information of a code. In classic coding theory, it gives the minimum distance of the code which determines the error correction capability of the code. In addition, the weight distribution allows the computation of the error probability of error detection and error correction with respect to some algorithms [2,16,31]. Thus, it is desirable to determine the weight distributions of linear codes. Moreover, linear codes with a few nonzero weights have many applications in constant composition codes [10], authentication codes [11] and secret sharing schemes [38] and some other fields. So it has provoked tremendous interests in determining the weight distributions of linear codes in literature. Different kinds of linear codes over finite fields and rings have been investigated explicitly for the past two decades, see [5,9,13,15,17,18,19,24,27,29,30,34,35,39]. In particular, Ding et al. [13] studied the weight distributions of a class of binary linear codes. Heng et al. dealt with projective binary linear codes from special Boolean functions in their recent work [18]. Huang et al. [19] constructed primitive binary LCD BCH codes and determined their parameters.

    Let q=pm for a prime p. Choose a subset D={d1,d2,,dn} of Fq, where Fq is the multiplicative group of Fq. Denote by Tr the absolute trace function from Fq to Fp. A linear code of length n is defined by

    CD={(Tr(bd1),Tr(bd2),,Tr(bdn)):bFq}. (1.1)

    The set D is called the defining set. Ding [12] pointed out that the defining-set construction is a fundamental approach and is equivalent to the generator matrix construction of all linear codes. Therefore it has attracted extensive attention and many families of linear codes were proposed following this way [1,13,14,21,22,23,33,36,37], most of which have good parameters. Particularly, Wu et al. [33] investigated three-weight binary linear codes from generalized Moisio's exponential sums. We refer the reader to [25,28] and the references therein for an overall survey on recent results and problems on constructions of linear codes from cryptographic functions.

    In the rest of the paper, we always take p=2 unless otherwise stated. In [13], a class of three-weight binary code CD of (1.1) is constructed using the defining set

    D={xFq:Tr(x2h+1)=0},

    where q=2m and 1h<m/2.

    Let α,βFq, and u a positive integer less than m. We consider a special case of the defining-set construction by defining a class of linear codes

    CD={c(a,b):a,bFq}, (1.2)

    where c(a,b)=(Tr(ax+by))(x,y)D and

    D={(x,y)F2q{(0,0)}:Tr(αx2u+1+βy2u+1)=0}. (1.3)

    The set D is also called the defining set of CD. Clearly, this is an extension of the work in [13].

    The purpose of this paper is to study the weight distributions of CD by employing Weil sums. These linear codes are projective with at most three nonzero weights and can be utilized to construct secret sharing schemes with good access structures.

    Now we present the main results of this paper and their proofs are given in Section 3. Let v=gcd(m,u) stand for the greatest common divisor of m and u. Let g be a generator of the cyclic group Fq. Namely, Fq=g. The weight distributions of CD are given in the following four theorems.

    Theorem 1.1. Let CD be defined by (1.2) and (1.3). If m/v is odd, then CD is a [22m11,2m,22m22m+v2] three-weight binary code with the weight distribution in Table 1.

    Table 1.  The weight distribution of CD in Theorem 1.1.
    Weight w Multiplicity Aw
    0 1
    22m22m+v2 2mv1(2mv+1)
    22m2 22m122m2v
    22m2+2m+v2 2mv1(2mv1)

     | Show Table
    DownLoad: CSV

    Theorem 1.2. Suppose that m/v is even and α,βg2v+1. Then CD is a [22m1+2m11,2m,22m2] two-weight binary code with the weight distribution in Table 2.

    Table 2.  The weight distribution of CD in Theorem 1.2.
    Weight w Multiplicity Aw
    0 1
    22m2 22m1+2m11
    22m2+2m1 22m12m1

     | Show Table
    DownLoad: CSV

    Theorem 1.3. Let m/v be even and α,βg2v+1. If um/2, then CD is a [22m1+2m+2v11,2m,22m2] three-weight binary code with the weight distribution in Table 3. If u=m/2, then CD is a simplex code with parameters [22m1,2m,22m1] and the only nonzero weight 22m1. Moreover, the simplex code meets the Griesmer bound.

    Table 3.  The weight distribution of CD in Theorem 1.3.
    Weight w Multiplicity Aw
    0 1
    22m2 2m2v1(2m2v+1)1
    22m2+2m+2v2 22m4v(24v1)
    22m2+2m+2v1 2m2v1(2m2v1)

     | Show Table
    DownLoad: CSV

    Theorem 1.4. Suppose that m/v is even and only one of α and β is in g2v+1, then CD is a [22m12m+v11,2m,22m22m+v1] three-weight binary code with the weight distribution in Table 4.

    Table 4.  The weight distribution of CD in Theorem 1.4.
    Weight w Multiplicity Aw
    0 1
    22m22m+v1 2mv1(2mv+1)
    22m22m+v2 22m22m2v
    22m2 2mv1(2mv1)1

     | Show Table
    DownLoad: CSV

    Some examples are provided to illustrate our main results. All of the numerical results are verified by Magma programs.

    Example 1. Let (m,u)=(3,1). By Theorem 1.1, the binary code CD has parameters [31, 6, 12]. Its weight enumerator is 1+10z12+47z16+6z20.

    Example 2. Let (m,u)=(2,1) and F4=g. If we take α=g2 and β=g, from Theorem 1.2 the binary code CD has parameters [9,4,4]. Its weight enumerator is 1+9z4+6z6. It is optimal according to Markus Grassl's code tables available at http://www.codetables.de/.

    Example 3. Let (m,u)=(4,2). Write F16=g and α=β=g5. By Theorem 1.3, the code CD has parameters [255,8,128] and it is an optimal simplex code with the only nonzero weight 128.

    Example 4. Let (m,u)=(4,1), F16=g, α=g3 and β=g. By Theorem 1.4, the code CD has parameters [111,8,48]. Its weight enumerator is 1+36z48+192z56+27z64.

    In this section, we present some results on group characters and Weil sums. Let G be a finite abelian group (written multiplicatively). A character χ of G is a homomorphism from G into the multiplicative group U of complex numbers of absolute value 1. That is, χ is a mapping from G into U with χ(xy)=χ(x)χ(y) for all x,yG. Let q=2m. For each bFq, the function

    χb(x)=(1)Tr(bx) for all   xFq

    defines an additive character of Fq, where Tr is the absolute trace function from Fq to F2. The additive character χ0 is called trivial, whereas other characters χb with bFq are called nontrivial. Especially χ1 is called the canonical additive character and is denoted by χ for simplicity. See [26] for more information about characters over finite fields.

    In [7], Coulter determined the value of Weil sums Su(α,β) defined by

    Su(α,β)=xFqχ(αx2u+1+βx),

    for all α,βFq, where q=2m and u is a positive integer. Recall that v=gcd(m,u) is the greatest common divisor of m and u.

    Lemma 2.1 (Theorem 4.1, [7]). If m/v is odd, then

    Su(α,0)={qif  α=0,0otherwise.

    Lemma 2.2 (Theorem 4.2, [7]). Let βFq and suppose m/v is odd. Then Su(α,β)=Su(1,βγ1), where γFq is the unique element satisfying γ2u+1=α. Further, we have

    Su(1,β)={0if  Trv(β)1,±2m+v2if  Trv(β)=1,

    where and hereafter Trv is the trace function from Fq to F2v.

    Lemma 2.3 (Theorem 5.2, [7]). Let m/v be even so that m=2k for some integer k. Then

    Su(α,0)={(1)k/v2kif  αgt(2v+1)  for  any  integer  t,(1)k/v2k+vif  α=gt(2v+1)  for  some  integer  t,

    where g is a generator of Fq.

    When m/v is even, the evaluation of Su(α,β) for p=2, where β0, was due to Coulter [7], and it can be similarly proved as the case of an odd prime p, see the poofs of Theorems 1 and 2 in [6].

    Lemma 2.4 (Theorem 5.3, [7]). Let βFq and suppose m/v is even such that m=2k for some integer k. Let fα(x)=α2ux22u+αxFq[x]. There are two cases.

    (i) If αgt(2v+1) for any integer t then fα is a permutation polynomial. Let x0Fq be the unique element satisfying fα(x0)=β2u. Then

    Su(α,β)=(1)k/v2kχ(αx2u+10).

    (ii) If α=gt(2v+1) for some integer t then Su(α,β)=0 unless the equation fα(x)=β2u is solvable. If the equation is solvable, with solution x0 say, then

    Su(α,β)=(1)k/v2k+vχ(αx2u+10).

    In this section, we always fix α,βFq and let g be a generator of Fq.

    The code length is defined by

    n=|D|=|{(x,y)F2q{(0,0)}:Tr(αx2u+1+βy2u+1)=0}|. (3.1)

    Lemma 3.1. The code length n of (3.1) is given as follows.

    (i) If m/v is odd, then n=22m11.

    (ii) If m/v is even, then

    n={22m1+2m11if  α,βg2v+1,22m1+2m+2v11if  α,βg2v+1,22m12m+v11otherwise.

    Proof. It follows from the orthogonal property of additive characters that

    n=12x,yFqz1F2(1)z1Tr(αx2u+1+βy2u+1)1=22m1+12x,yFq(1)Tr(αx2u+1+βy2u+1)1=22m11+12Su(α,0)Su(β,0).

    Thus we obtain the desired conclusions from Lemmas 2.1 and 2.3.

    The Pless power moments are useful tools when we calculate the weight distribution of a given code. Recall that the code CD is defined by (1.2) and (1.3) with length n and dimension κ=dimF2(CD). The weight distributions of CD and its dual CD are denoted by (1,A1,,An) and (1,A1,,An), respectively. As we will prove later in Theorem 4.1, the minimum weight of the dual code CD is at least 3. So A1=0, A2=0 and consequently the first three Pless power moments are given by [20Y, p.260]:

    nj=0Aj=2κ,nj=0jAj=2κ1n,nj=0j2Aj=2κ2n(n+1).

    In this subsection, we will prove the weight distributions of CD given in Theorems 1.1, 1.2, 1.3 and 1.4. The code length n is given in Lemma 3.1. Assume that (a,b)(0,0) unless otherwise stated. We define

    N0(a,b)=|{(x,y)F2q:Tr(αx2u+1+βy2u+1)=0,Tr(ax+by)=0}|. (3.2)

    Then the Hamming weight of c(a,b) is expressed as

    wt(c(a,b))=nN0(a,b)+1. (3.3)

    By (3.2) and the orthogonal property of additive characters,

    N0(a,b)=22x,yFqz1F2(1)z1Tr(αx2u+1+βy2u+1)z2F2(1)z2Tr(ax+by)=22x,yFq(1+(1)Tr(αx2u+1+βy2u+1))(1+(1)Tr(ax+by))=22m2+22(Su(α,0)Su(β,0)+Su(α,a)Su(β,b)). (3.4)

    Now we are going to determine the values of N0(a,b) given by (3.4). There are four cases to consider according to the parity of m/v and the values of α and β.

    In the first case, if m/v is odd, the length is n=22m11. At first glance, when a=0 and b0, we have Su(α,0)=0 by Lemma 2.1. So N0(a,b)=22m2. Similarly when a0 and b=0, N0(a,b)=22m2. Assume that aFq, we have from Lemma 2.2 that

    Su(α,a)=Su(1,aγ1)={0 if Trv(aγ1)1,±2m+v2 if Trv(aγ1)=1,

    where γFq is the unique element satisfying γ2u+1=α. Thus it follows from (3.4), Lemmas 2.1 and 2.2 that

    N0(a,b){22m2,22m2+2m+v2,22m22m+v2}.

    Hence, by (3.3), the weight wt(c(a,b)) of the codeword c(a,b) satisfies

    wt(c(a,b)){22m2,22m2+2m+v2,22m22m+v2}.

    Put

    w1=22m22m+v2,w2=22m2,w3=22m2+2m+v2.

    We now determine the number Awi of codewords with weight wi in CD. The first three Pless power moments yield the following system of equations:

    {Aw1+Aw2+Aw3=22m1,w1Aw1+w2Aw2+w3Aw3=22m1n,w21Aw1+w22Aw2+w23Aw3=22m2n(n+1), (3.5)

    where n=22m11. Solving the system of equations in (3.5) leads to the weight distribution given in Table 1. This proves Theorem 1.1.

    In the second case, if m/v is even and α,βg2v+1, the length is n=22m1+2m11. It follows from Lemmas 2.3 and 2.4 that

    Su(α,0)=(1)k/v2k,Su(α,a)=(1)k/v2kχ(αx2u+10),

    where a0 and x0 satisfies fα(x0)=a2u. By (3.4),

    N0(a,b){22m2,22m2+2m1}.

    From (3.3), the weight wt(c(a,b)) belongs to the set

    {22m2,22m2+2m1}.

    Let

    w1=22m2,w2=22m2+2m1.

    Again by solving the system of equations

    {Aw1+Aw2=22m1,w1Aw1+w2Aw2=22m1n, (3.6)

    where n=22m1+2m11, we get the weight distribution given in Table 2. This finishes the proof of Theorem 1.2.

    In the third case, if m/v is even and α,βg2v+1, the length is n=22m1+2m+2v11. Again from Lemma 2.3, we have

    Su(α,0)=(1)k/v2k+v.

    Let a0.

    It follows from Lemma 2.4 that Su(α,a)=0 or if the equation fα(x)=a2u is solvable with a solution x0Fq, then

    Su(α,a)=(1)k/v2k+vχ(αx2u+10).

    By (3.3) and (3.4), the weight wt(c(a,b)) belongs to the set

    {22m2,22m2+2m+2v2,22m2+2m+2v1}.

    Write

    w1=22m2,w2=22m2+2m+2v2,w3=22m2+2m+2v1.

    The first three Pless power moments are given by (3.5), where n=22m1+2m+2v11. Solving these equations yields the weight distribution given in Table 3. This completes the proof of Theorem 1.3.

    The last case is that m/v is even and αg2v+1, βg2v+1 (or βg2v+1, αg2v+1). In this case, n=22m12m+v11. After a similar argument as we have done in the previous case, we obtain from (3.3) and (3.4) that wt(c(a,b)) belongs to the set

    {22m2,22m22m+v2,22m22m+v1}.

    Set

    w1=22m22m+v1,w2=22m22m+v2,w3=22m2.

    From the first three Pless power moments (3.5), we get the weight distribution given in Table 4, completing the proof of Theorem 1.4.

    For the dual CD of the code CD, we have the following conclusion.

    Theorem 4.1. Let m2 and α,βFq. The dual CD of the code CD is a binary code with parameters [n,n2m,d], where n is given in Lemma 3.1 and d=3 if m is even and 3d4 if m is odd.

    Proof. The dimension of the code CD is obvious. Since D does not contain the zero element of F2q, the minimum distance of CD cannot be one. Similarly, since D is not a multiset, any two elements di and dj of D must be distinct if ij. Hence, the minimum distance CD cannot be 2. So we have d3.

    When m is even, we assume that (x1,0),(0,y2)D. We claim that (x1,y2) is in D. Actually,

    Tr(αx2u+11+βy2u+12)=Tr(αx2u+11)+Tr(βy2u+12)=0.

    Therefore, the minimum distance of CD is 3.

    When m is odd, n=22m11 by Theorem 1.1. Let D={di=(d1i,d2i):i=1,2,,n}. Consider the sums di+dj for ij. The total number of such sums is equal to (22m11)(22m21)>22m for m2. Hence, there must be four distinct integers i,j,k,l{1,2,,n} such that di+dj=dk+dl. This means that CD has a codeword with Hamming weight 4. So we have 3d4, completing the whole proof.

    When m is odd, the code CD is at least almost optimal. This is because the minimum weight of any binary code with length 22m11 and dimension 22m112m is at most 4 according to the sphere packing bound.

    Example 5. Let (m,u)=(2,1), α=g2 and β=g, where F4=g. Magma programs show that the binary code CD has parameters [9,5,3] and it is optimal. If we take α=β=g3, then CD has parameters $ [15,11,3] and it is optimal, too.

    Example 6. Let (m,u)=(3,1). Then the binary code CD has parameters [31,25,3] and it is almost optimal, while the optimal binary code has parameters [31,25,4].

    In this paper, a class of projective binary codes with two or three weights were constructed from a proper defining set. Their weight distributions were determined by applying Weil sums and the first three Pless power moments. Furthermore, we determined the parameters of their dual codes. Some optimal and almost optimal codes were also constructed. Due to [38], a linear code over F2 is suitable to construct secret sharing schemes with interesting access structures if

    wminwmax>12, (5.1)

    where wmin and wmax denote the minimum and maximum nonzero weights of the code, respectively. For the linear codes CD in Theorems 1.1-1.4, the inequality (5.1) always holds if m2v+2. So they can be used in secret sharing schemes with good access structures. Additionally, projective two-weight codes in Theorem 1.2 can be applied in strongly regular graphs [4,8] and projective three-weight codes in Theorems 1.1, 1.3 and 1.4 are related to association schemes with three classes [3].

    The authors would like to thank the referee for his/her thorough review with constructive suggestions and valuable comments. The work is partly supported by the National Natural Science Foundation of China under Grant 12071247, Grant 11701317, Grant U1811461 and Grant 11971496.

    The authors declare that there is no conflict of interests regarding the publication of this paper.



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