Research article

Weight distributions of a class of skew cyclic codes over M2(F2)

  • Received: 25 February 2025 Revised: 08 May 2025 Accepted: 14 May 2025 Published: 20 May 2025
  • MSC : 11T71, 94B05, 94B15

  • Let M2(F2) be the ring of matrices of order 2×2 over finite field F2 and ωM2(F2) be a cubic primitive root of unity. For any even positive integer t, the weight distributions of the skew cyclic codes of length 3t with parity check polynomials xtωi,i=0,1,2 and (xtωj)(xtωk), 0j<k2 were determined.

    Citation: Zhen Du, Chuanze Niu. Weight distributions of a class of skew cyclic codes over M2(F2)[J]. AIMS Mathematics, 2025, 10(5): 11435-11443. doi: 10.3934/math.2025520

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  • Let M2(F2) be the ring of matrices of order 2×2 over finite field F2 and ωM2(F2) be a cubic primitive root of unity. For any even positive integer t, the weight distributions of the skew cyclic codes of length 3t with parity check polynomials xtωi,i=0,1,2 and (xtωj)(xtωk), 0j<k2 were determined.



    Let M2(F2) be the full matrix ring over finite field F2 of order 2×2 and C be a linear code over M2(F2) with parameters [n,k,d], where n is the length, k is the dimension, and d is the minimum distance. For an automorphism θ of M2(F2) and a unit λ in M2(F2), the linear code C is said to be a skew constacyclic code or a (θ,λ)- constacyclic code if it is closed under the (θ,λ) shift κθ,λ defined by

    κθ,λ((c0,c1,,cn1))=(θ(λcn1),θ(c0),,θ(cn2)).

    In particular, such codes are called skew cyclic or skew negacyclic codes when λ equals 1 or 1, respectively. Specifically, when θ is the identity automorphism, then skew constacyclic code C becomes a classical constacyclic, cyclic, and negacyclic code.

    The weight distribution of C is defined by the sequence (1=A0,A1,A2,,An), where Ai is the number of codewords with Hamming weight i of C. The weight enumerator of C is defined by

    1+A1z+A2z2++Anzn.

    The set Ω={i:Ai0} is called its weight set. In coding theory, the weight distribution of codes plays an important role, as it can help us analyze the minimum distance between codewords and to understand the error correction performance. By analyzing the weight distribution of the code, the error correction ability and decoding performance of the code can be improved.

    The weight distributions of cyclic codes have been studied for many years and are well-known in some cases. There are extensive studies on the weight distributions of reducible and irreducible cyclic codes. The weight distribution of irreducible cyclic codes over finite fields has been thoroughly investigated in [9,21,25], and the weight distribution of reducible cyclic codes over finite fields can be found in [5,22,24]. The weight distribution of cyclic codes with few weights is studied in [12,13,20]. The weight distribution of fixed-length cyclic codes was studied in [10,15,26]. The complete weight distribution of two classes of cyclic codes was studied in [7]. Furthermore, there are significant findings and further researches on weight distributions of cyclic codes contained in [2,8,11].

    In the history of error-correcting code theory, skew cyclic codes have also attracted significant attention. D. Boucher conducted initial studies of skew cyclic codes in [3], and D. Boucher also studied skew cyclic codes over Galois rings in [4]. S.Jitman studied skew constacyclic codes over finite chain rings in [6]. M. Shi studied skew cyclic codes over non-chain ring in [17], and I. Siap studied skew cyclic codes of arbitrary lengths in [19]. In addition, there are also studies on skew cyclic codes over chain rings (see [1,14,16]).

    Based on the algebraic structures of skew cyclic codes over M2(F2) obtained by [18], we study the weight distributions of a class of skew cyclic codes over M2(F2). Let F4 be the splitting field of f(x)=x2+x+1 and ω be a root of f(x) in F4. For any even positive integer t, the weight distributions of the skew cyclic codes of length 3t with parity check polynomials xtωi,i=0,1,2 and (xtωj)(xtωk), 0j<k2 are determined. The major results are presented in the following theorems.

    Theorem 1.1. Let C be the skew cyclic code with parity check polynomials h(x)=xtωi,i=0,1,2. The weight distribution of C is the sequence (A0,A1,A2,,An), where

    Ai={0,3i;(ti/3)15i/3,3i.

    Theorem 1.2. Let C be the skew cyclic code with parity check polynomials (xtωj)(xtωk), 0j<k2. Then, the weight set of C is Ω={3i+2j|0it,0jti}, and the weight distribution is

    Ak=ti=0tij=03i+2j=k(ti)(tij)210i45j,k=0,1,,n.

    This paper is organized as follows. We review the algebraic structure of skew cyclic codes over M2(F2) briefly in Section 2. In Section 3, we prove our major results and give some examples. In Section 4, we concludes this paper.

    We review some algebraic properties of skew cyclic codes over M2(F2) briefly in this section.

    We denote the 2×2 matrix ring over finite field F2 by M2(F2). By [23, p. 111], the matrix ring M2(F2) is isomorphic to the F4-cyclic algebra R=F4eF4 with e2=1 under map δ,

    δ:(0110)e,(0111)ω.

    Denote σ the Frobenius map on F4. The Frobenius σ can be extended to a homomorphism θ on R that is defined by θ(a+eb)=σ(a)+eσ(b), a,bF4. The multiplication in R is given by re=eθ(r) for any rR, and the addition is usual.

    A linear code C on R is a left submodule of Rn. The linear code C is said to be l-quasi cyclic if it is closed under Tl, where T is the linear shift mapping defined by

    T(c0,c1,,cn1)=(cn1,c0,,cn2).

    In particular, the linear code C is a cyclic code if l=1. The linear code C is called a skew cyclic code if it is closed under the map Tθ, defined by

    Tθ(c0,c1,,cn1)=(θ(cn1),θ(c0),,θ(cn2)).

    Denote

    R[x,θ]={rnxn+rn1xn1++r1x+r0|riR,0in,nN}.

    For any rR and any natural number i, define the multiplication by xir=θi(r)xi. Then, for any a+ebR, we have

    xi(a+eb)={(a+eb)xi,if i is even,(a2+eb2)xi,if i is odd.

    We can see that R[x,θ] forms a skew polynomial ring under this multiplication and the usual polynomial addition. We also denote R[x,θ]t as the subset of R[x,θ] of polynomial of degree less than t.

    Lemma 2.1. [18, Lemma 2.1] Let f(x), g(x)R[x,θ] with the leading coefficient of g(x) as invertible. Then, there exist two unique polynomials q(x), r(x)R[x,θ], such that

    f(x)=q(x)g(x)+r(x),

    with r(x)=0 or deg(r(x))<deg(g(x)). The polynomials q(x) and r(x) are called the right quotient and right remainder, respectively. The polynomial g(x) is called a right divisor of f(x) if r(x)=0.

    Lemma 2.2. The center Z(R[x,θ]) of a polynomial ring R[x,θ] is F2[x2].

    Proof. For any rR, we have x2ir=(θ2)i(r)x2i=rx2i. Thus, x2iZ(R[x,θ]). This implies that if rjF2, then sj=0rjx2j lies in Z(R[x,θ]). Conversely, for any fzZ(R[x,θ]) and rR, if rfz=fzr and xfz=fzx, then the coefficients of fz are in F2 and fzR[x2,θ]. Therefore, fzF2[x2].

    Lemma 2.3. [18, Proposition 2.6.] Let n be a positive integer and C be a skew cyclic code of length n over R with a polynomial of minimum degree d(x), where the leading coefficient of d(x) is a unit. Then, C is a free R[x,θ]-submodule. R[x]-submodule of Rn, such that C=<d(x)>, where d(x) is a right divisor of xn1. Moreover, the code C has a basis B={d(x),xd(x),...,xndeg(d(x))1d(x)}, and the number of codewords in C is |R|ndeg(d(x)).

    Let t be an even integer and n=3t. Denote Rn=R[x,θ]/xn1. By Lemma 2.2, the sets Rn=R[x,θ]/xn1 and R[x,θ]/xt1 are two rings. It is easy to check xt1, xtω and xtω2 commute with each other, so in R[x,θ], we have

    xn1=x3t1=(xt1)(xtω)(xtω2).

    Under the canonical projection ψ, it is known by Chinese remainder theorem, as left R[x,θ]-modules,

    R[x,θ]/xn1R[x,θ]/xt1R[x,θ]/xtωR[x,θ]/xtω2. (2.1)

    In fact, it is easy to prove the injection and the module homomorphism, whereas the surjection follows since we can define

    f(x)=f1(x)(x2t+xt+1)+f2(x)(ωx2t+ω2xt+1)+f3(x)(ω2x2t+ωxt+1)

    for any (¯f1(x),¯f2(x),¯f3(x))R[x,θ]/xt1R[x,θ]/xtωR[x,θ]/xtω2.

    In this section, let t be an even integer and n=3t. The code C is the skew cyclic code with parity check polynomial h(x) of the form xtωi,i=0,1,2 or (xtωj)(xtωk), 0j<k2. We determine the weight distributions of C and prove the major theorems.

    For any ϕ(x)R[x,θ]/xn1, there exists unique (r0(x),r1(x),r2(x))(R[x,θ]t)3 such that ϕ(x)=2i=0ri(x)xti, therefore,

    ψ(ϕ(x))=(b0(x),b1(x),b2(x)), (3.1)

    where bj(x)=2i=0ri(x)(ωi)j,j=0,1,2. Denote

    M=[1111ωω21ω2ω4]=[1111ω1+ω11+ωω],

    then

    ψ(ϕ(x))=(b0(x),b1(x),b2(x))=(r0(x),r1(x),r2(x))M. (3.2)

    Therefore, we have

    (r0(x),r1(x),r2(x))=(b0(x),b1(x),b2(x))M1, (3.3)

    where

    M1=[11111+ωω1ω1+ω].

    As the proofs are similar, we assume h(x)=xt1. Then under the canonical projection ψ defined by (2.1),

    CR[x,θ]/xt1,

    and for any 2i=0rixtiC, we have ψ(2i=0rixti)=(b0(x),0,0). Hence, by (3.3), we have

    b0(x)=r0(x)=r1(x)=r2(x). (3.4)

    Theorem 3.1. The weight distribution of C is the sequence (A0,A1,A2,,An), where

    Ai={0,3i;(ti/3)15i/3,3i.

    Proof. Let s be the number of nonzero coefficients of b0(x), then by (3.4), the number of nonzero coefficients of r0(x), r1(x) and r2(x) is 3s. Hence, the weight set of C is Ω={3s0st}. Furthermore, since there are (ti)15i polynomials b0(x) in R[x,θ]t of weight i, Theorem 3.1 is proved.

    Example 1. Here are two examples:

    (1) When t=8, then C is a [24,8,3] skew cyclic code over M2(F2) with weight enumerator

    1+120z3+6300z6+18900z9+3543750z12+42525000z15+318937500z18+1366875000z21+2562890625z24.

    (2) When t=10, then C is a [30,10,3] skew cyclic code over M2(F2) with weight enumerator

    1+150z3+10125z6+405000z9+10631250z12+191362500z15+2392031250z18+20503125000z21+115330078125z24+384433593750z27+576650390625z30.

    We assume h(x)=(xt1)(xtω) as the proofs of other cases are similar. Then, then under the canonical projection ψ defined by (2.1),

    CR[x,θ]/xt1R[x,θ]/xtω

    and for any 2i=0ri(x)xtiC, we have ψ(2i=0ri(x)xti)=(b0(x),b1(x),0). By (3.3), we get

    {b0(x)+b1(x)=r0(x),b0(x)+(1+ω)b1(x)=r1(x),b0(x)+ωb1(x)=r2(x). (3.5)

    For i=0,1,2, write bi(x)=t1s=0bi,sxs and ri(x)=t1s=0ri,sxs. By comparing the coefficients on both sides of (3.4), we obtain t system of equations as follows:

    {b0s+b1s=r0s,b0s+(1+ω)b1s=r1s,0st1.b0s+ωb1s=r2s, (3.6)

    Since any two vectors of (1,1), (1,ω) and (1,1+ω) are linearly independent, then for any (b0s,b1s)R2,(3.6) implies the number of nonzero entries of (r0s,r1s,r2s)R3 cannot be 1; therefore, the Hamming weight wH(r0s,r1s,r2s) lies in S={0,2,3}. Hence, the Hamming weight set of C is

    Ω={kk=k0+k1++kt1,kiS,0it1}.

    By counting the number of those ki's that equal 2 and 3, we have

    Ω={3i+2j0it,0jti}. (3.7)

    From the Hamming weight set Ω obtained in (3.7), it can be seen that C is a [n=3t,2t,2] code and Theorem 1.2 follows from Theorem 3.2.

    Theorem 3.2. The weight distribution of C is

    Ak=ti=0tij=03i+2j=k(ti)(tij)210i45j,k=0,1,,n.

    Proof. Considering the following system of equations

    {x1+x2=0,x1+(1+ω)x2=0,x1+ωx2=0. (3.8)

    For 0k3, let M(k) be the number of the points (x1,x2)R2 that satisfies some k equations of (3.8), but does not satisfy the other, and let N(k) be the number of (b0s,b1s)R2 such that the corresponding Hamming weight of (r0s,r1s,r2s)R3 equals k, then N(k)=M(3k).

    Since any two equations of (3.8) have only common zero solution, then M(2)=0 and M(3)=1. For each equation, there are 15 nonzero solutions, and therefore M(1)=3×15=45, which implies that M(0)=162145=210. In conclusion, values of M(k)s and N(k)s are listed in Tables 1 and 2.

    Table 1.  Values of M(k)s.
    k 0 1 2 3
    M(k) 210 45 0 1

     | Show Table
    DownLoad: CSV
    Table 2.  Values of N(k)s.
    k 0 1 2 3
    N(k) 1 0 45 210

     | Show Table
    DownLoad: CSV

    For any 0kn=3t, the number Ak of codewords 2i=0ri(x)xti of weight k in C can be obtained by collecting the number of all those (b0s,b1s)R2, such that wH(r0s,r1s,r2s)=ks with k=k0+k1++kt1, then

    Ak=(k0,,kt1)Stk=k0++kt1N(k0)N(k1)N(kt1)=ti=0tij=03i+2j=k(ti)(tij)210i45j.

    Theorem 3.2 can be proved.

    Example 2. We list two examples:

    (1) Let t=2, then C is a [6,4,2] skew cyclic code over M2(F2) with weight enumerator

    1+90z2+420z3+2025z4+18900z5+44100z6.

    (2) Let t=4, then C is a [12,8,2] skew cyclic code over M2(F2) with weight enumerator

    1+180z2+840z3+12150z4+113400z5+629100z6+5103000z7+27914625z8+113589000z9+535815000z10+1666980000z11+1944810000z12.

    In this paper, for any even positive integer t, we present two major theorems for calculating the weight distribution of a class of skew cyclic codes of length 3t. It is interesting to study case t, as it is odd.

    Zhen Du: Conceptualization, Writing-original draft; Chuanze Niu: Conceptualization, Writing-review and editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We thank the editors and the reviewers for their useful feedback that improved this paper. This work is supported by the Natural Science Foundation of Shandong Province, China (Grant number ZR2019BA011) and by the National Natural Science Foundation of China (Grant number 11401285).

    The authors declare no conflicts of interest.



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