Weight distribution of a class of cyclic codes of length 2 n Research

Let Fq be a finite field with q elements and n be a positive integer. In this paper, we determine the weight distribution of a class cyclic codes of length 2 over Fq whose parity check polynomials are either binomials or trinomials with 2 zeros over Fq, where integer l ≥ 1. In addition, constant weight and two-weight linear codes are constructed when q ≡ 3 (mod 4). 2010 MSC: 12E05, 12E20, 94B05, 11T71


Introduction
A linear [m, k] q code C over F q is a k-dimensional subspace of F m q .Let λ be a nonzero element in F q .A linear code C of length m over F q is called a λ-constacyclic code if (c 1 , c 2 , . . ., c m−1 , c 0 λ) ∈ C, ∀ (c 0 , c 1 , c 2 , . . ., c m−1 ) ∈ C.
A λ-constacyclic code of length m over F q is called a simple-root code if gcd(m, q) = 1; otherwise it is called repeated-root code.If we identify a vector (a 0 , a 1 , . . ., a m−1 ) ∈ F m q with a polynomial a 0 ], then a simple-root λ-constacyclic code C can be interpreted as an ideal of the quotient ring R = F q [x]/ x m − λ .It is well known that each ideal in R is of the form g(x) , where g(x) is a monic divisor of x m − λ in F q [x].The polynomial g(x) is known as the generator polynomial of the code C and the corresponding factor h(x) = (x m − λ)/g(x) of x m − λ is referred to as the parity check polynomial of the code C. If h(x) is a product of t monic irreducible factors over F q , then we say C with t zeros over F q .A constacyclic code is called an irreducible code over F q if t = 1, Manjit Singh (Corresponding Author), Sudhir Batra; Department of Mathematics, Deenbandhu Chhotu Ram University of Science and Technology, Murthal-131039, Sonepat, India (email: manjitsingh.math@gmail.com,batrasudhir@rediffmail.com).and a reducible code over F q if t ≥ 2. A λ−constacyclic code C is called cyclic if λ = 1 and negacyclic if λ = −1.
Let A i be the number of codewords of a linear code C of length m over F q with Hamming weight i, where 0 ≤ i ≤ m.Note that A 0 = 1 and A i = 0 for all 1 ≤ i < d, where d is the minimum Hamming distance of the code.The sequence (1, A 1 , A 2 , . . ., A m ) is called the weight distribution of the code C. Cyclic codes are the most important class of linear codes for a wide variety of applications.In the last few decades, the weight distribution of irreducible cyclic codes have been studied extensively (see [3,9,12]).However, not much is known about the weight distribution of reducible codes except in very specific cases.
Vega [11] presented a new family of two-weight reducible cyclic codes that can be constructed as the direct sum of two one-weight irreducible codes.For any q = p m , where p is an odd prime, m ≥ 3, k ≥ 1 and gcd(k, m) = 1, Feng and Luo [4] obtained weight distribution of cyclic codes C 1 of dimension 2m and C 2 of dimension 3m over F p of length n = q − 1 with parity-check polynomial h 2 (x)h 3 (x) and h(x) = h 1 (x)h 2 (x)h 3 (x) respectively, where h 1 (x), h 2 (x) and h 3 (x) are the minimal polynomials of π −1 , π −2 and π −(p k +1) over F p , respectively, for a primitive element π of F q with degh i (x) = m for i = 1, 2, 3.For doing this, they computed the value distribution of multi-sets of exponential sums using quadratic form over F p .
Yang, Xiong, Ding and Luo [14] proposed a class of cyclic codes C of length n over F q with paritycheck polynomial h(x) = h a1 (x)h a2 (x) • • • h at (x), where h ai (x) is the minimal polynomial of γ −ai over F q , degh ai (x) = m, r = q m , γ is a generator of F * r and n = (r − 1)/δ, δ = gcd(r − 1, a 1 , a 2 , . . ., a t ) and integer e ≥ t ≥ 2 with e|(q − 1).They remarked that it may be very difficult to find the weight distribution of this class of codes if the integers a i are not chosen in the right way or the Gaussian periods have many different values.The values of the Gaussian periods are in general very hard to determine.Hence they obtained the weight distribution of this class of codes when t = e and the Gaussian periods of order N are known, including the cases that N = 1, 2, 3, semi-primitive case and a special index 2 case.
Recently, assuming v ||(q − 1), where is a prime and v is an integer, and q ≡ 1 (mod 4) if = 2, Zhu, Yue and Hu [15] have applied a combinatorial method to obtain not only the weight distribution of all cyclic codes of length m with two zeros over F q , but also the weight distribution of a special cyclic code of length m with three zeros over F q .
In this paper, we present a class of cyclic codes of length 2 n with 2 l zeros over F q , where q is an odd prime power and n > l ≥ 1.Further, using the explicit forms of codewords, the weight distribution of these codes is determined explicitly.We make use of Diophantine equation and its solutions to obtain the explicit form of weights of codewords and the number of codewords of the given weight of these cyclic codes.It is observed that, when q ≡ 3 (mod 4), the problem of finding the weight distribution is transferred into a problem of determining the weight distribution of a two-weight negacyclic code, which turns out to be associated with counting the number of constant weight linear codes.These codes are known for applicability of various association schemes and traceability schemes, which justify their practical applications in engineering perspective (see [1,2,[5][6][7]).In particular, these codes are of special interest in authentication codes [2] and traceability schemes [6].
The paper is organized as follows: The necessary notations and some auxiliary results are provided in Section 2. In Section 3, we describe a class of negacyclic and hence cyclic codes of length 2 n with 2 l zeros over F q .It is observed that these reducible codes are reversible when q ≡ 3 (mod 4).In Section 4, constant weight linear codes and two-weight negacyclic codes are constructed and the weight distribution of cyclic codes of length 2 n with 2 l zeros over F q are determined.In the end of this section, we give an example illustrating the results.In Section 5, we conclude the paper.

Preliminaries
The paper follows the standard notation of finite fields.The multiplicative group of F q is denoted by F * q .It is well known that F * q is a cyclic group of order q − 1.For any integer m ≥ 2, let ν 2 (m) denote the highest power of 2 dividing m.Let q be an odd prime power, s = ν 2 (q − 1) and u = ν 2 (q 2 − 1).Then u − s = ν 2 (q + 1) ≥ 1. Readily note that (i) u − s = 1 if and only if q ≡ 1 (mod 4), and (ii) s = 1 if and only if q ≡ 3 (mod 4).
Let α k be a primitive 2 k th root of unity in F * q .Also, let β k be a primitive 2 k th root of unity in F * If f (x) is an irreducible polynomial over F q and f (0) = 0, then, for any integer k ≥ 2, the following result is useful to check the irreducibility of the polynomial of the type of f (x k ) over F q .
Lemma 2.1.[13, Theorem 10.15] Let f (x) be any irreducible polynomial over F q of degree l ≥ 1. Suppose that f (0) = 0 and f (x) is of order e which is equal to the order of any root of f (x).Let k be a positive integer, then the polynomial f (x k ) is irreducible over F q of order ke if and only if the following three conditions are satisfied: (i) Every prime divisor of k divides e; (ii) gcd(k, q l −1 e ) = 1; (iii) If 4|k, then 4|(q l − 1).
We end this section with the following lemma.Lemma 2.2.Let q be an odd prime power and r ≥ 1 be an integer.Then (i) For any 2 ≤ k ≤ s, x 2 r − α k is a product of 2 l monic irreducible binomial factors of degree 2 r−l over F q , where l = min{r, s − k}.
(ii) For any s k is a product of 2 l monic irreducible trinomial factors of degree 2 r−l over F q , where l = min{r, u − k}.
Proof.(i) For any fixed 2 ≤ k ≤ s, let l = min{r, s − k}.Then 2 k+l |(q − 1), α k+l ∈ F * q such that α k = α 2 l k+l .The set In view of the part (i), x 2 r−l − η is an irreducible factor of x 2 r − β k over F q 2 for every η ∈ T k .For any η ∈ T k , we notice that β k = η 2 l and β q k = (η q ) 2 l .Further, x 2 r−l − η is a factor of x 2 r − β k over F q 2 if and only if x 2 r−l − η q is a factor of x 2 r − β q k over F q 2 .This bijection of factors of

Negacyclic and cyclic codes
For any 1 ≤ k ≤ s and integer r ≥ 0, we define a negacyclic [2 k+r−1 , 2 r ] q code and a cyclic [2 k+r , 2 r ] q code over F q by N r (x) , respectively, with the generator polynomials Note that N (1) r is the whole space F 2 r q .By Lemma 2.2, for each 2 ≤ k ≤ s, N (k) r and C (k) r are the codes with 2 l zeros over F q , where l = min{r, s − k}.Lemma 3.1.For any 1 ≤ k ≤ s and integer r ≥ 0, and for any r ≥ 0, Therefore the generator polynomial of .
Let b = (b 0 , b 1 , . . ., b 2 r −1 ) ∈ F 2 r q be a message word and the corresponding message polynomial be b Then the code polynomial of N (k) r is given by: b(x) x The polynomial on the right hand side can be expressed as a vector of the form (b, bα k , . . ., bα 2 k−1 −1 k ) by substituting i = 0, 1, . . ., 2 k−1 − 1.Hence, we have

Since the generator polynomial of C
Lemma 3.1 is valid for every odd q.Consider the case q ≡ 3 (mod 4), i.e., s = 1, then we obtain the following trivial codes: In the rest of this section, we assume q ≡ 3 (mod 4).In order to define a class of cyclic codes with parity check polynomial of the type x 2 r+1 − (β k + β q k )x 2 r + β q+1 k with 2 l zeros, where l = min{r, u − k}, we need a bit more notations of [10].
For any x ∈ F q 2 \ F q , let T(x) = x + x q and N(x) = x q+1 .It follows that the minimal polynomial of β 2 over F q is x 2 + 1, and the minimal polynomial of β k over F q is x 2 − T(β k )x + N(β k ) for every 3 ≤ k ≤ u.For every 1 ≤ i ≤ 2 k−2 and 3 ≤ k ≤ u, there are exactly 2 k−2 elements of the form T( In order to avoid cumbersome notations, for a fixed k, we denote ξ = T(β k ), = β q+1 k .Note that ξ = 0 if k = 2; = 1 for 2 ≤ k ≤ u − 1 and = −1 for k = u.Further, for any integer i ≥ 0 and 3 ≤ k ≤ u, we define the followings: Lemma 3.2.Let 3 ≤ k ≤ u be a fixed integer and a, b ∈ F q .Then, the sequence (γ i (a, b)) i≥0 satisfies the recursive relation for any integer r ≥ 0, For any integer r ≥ 0, we define a negacyclic ) is said to be reversible provided f * (x) = f (x).A reversible code is a code such that reversing the order of the components of a codeword gives always a codeword.Massey [8] showed that reversible cyclic codes are those which have self-reciprocal generator polynomials.For any integer r ≥ 0, the polynomial x 2 r+1 − ξx 2 r + 1 is reversible.By Lemma 2.2, x 2 r+1 − ξx 2 r + 1 is reducible over F q and hence C r,k is a reducible and reversible cyclic code.Cyclic codes can be decoded by sequential circuits, and hence the invariance of these codes under the reversing transformation is of special interest [8].
The code polynomial of N k is In view of Lemma 3.2, we obtain the desired form of N k .Now, for any integer r ≥ 0, we have a j x 2 r+1 −j−1 be the polynomial representation of the vector a = (a 0 , a 1 , . . ., a The code polynomial of N r,k is Then any codeword of N r,k can be expressed as (c 0 , c 1 , . . ., c 2 r+k−1 −1 ) in which for every This completes the proof.

Concluding remarks
The main contributions of this paper are the followings: • The construction of a class of linear codes of length 2 n with 2 l zeros over F q and their weight distribution.These codes are reversible when l ≥ 1 and q ≡ 3 (mod 4).
• The explicit form of the weight distribution in which the weights of codewords and the number of codewords of a given weight of these codes can be computed easily using a linear Diophantine equation and its solutions (see Example 4.8).
• The construction of constant weight linear codes and two-weight negacyclic codes of length 2 n , where 2 n divides q + 1 and integer n ≥ 2. A class of linear codes with few weights are of special interest in authentication codes [2] and traceability schemes [6].
Many authors have worked on the problem of determining the weight distribution of reducible cyclic codes using mathematical tools, such as Gaussian periods and exponential sums.The values of the Gaussian periods, exponential sums are in general very hard to compute.It would be interesting to use the combinatorics approach of this paper for obtaining the weight distribution of cyclic codes of length m over F q whose parity check polynomials are binomials or trinomials over F q for the case m ∈ {2 n d, d n } for some odd integer d such that d|(q − 1) or d|(q + 1).

Table 4 .
8 characterizes the type of possible weights = 8x + 6y of a reducible cyclic [16, 4, 6] 7 code with the number of codewords A of a given weight .