Hamming weight enumerators of multi-twisted codes with at most two non-zero constituents
Introduction
Let denote the finite field of order q, n be a positive integer, and let denote the vector space consisting of all n-tuples over . A linear code of length n over is defined as an -linear subspace of , whose dimension k is called the dimension of the code . Elements of the code are called codewords, and the Hamming weight of a codeword of is defined as the number of its non-zero components. The Hamming distance d of the code is the smallest of the Hamming weights of its non-zero codewords. The greater is the Hamming distance of a code, higher are its error-detecting and error-correcting capabilities. A linear code of length n, dimension k and Hamming distance d over is referred to as a linear -code over . The Griesmer bound is a lower bound on the length n of the code for given dimension k and Hamming distance d, while the Plotkin bound is an upper bound on the size of the code and is applicable only when . Linear codes attaining either the Griesmer bound or the Plotkin bound are optimal linear codes and have attracted the attention of many coding theorists [16], [18], [19], [21]. Besides the length n, dimension k and Hamming distance d, another important parameter of the code is its Hamming weight distribution, which is defined as the list , where denotes the number of codewords in having the Hamming weight j for . The polynomial is called the Hamming weight enumerator of the code . The Hamming weight enumerator (or equivalently, the Hamming weight distribution) of a code is useful in studying its error-performance relative to various communication channels [5], [10], [28]. Thus the problem of determination of the Hamming weight enumerator (or equivalently, the Hamming weight distribution) of a code is of great interest [11], [13], [14], [16], [19], [21], [24], [25], [29]. In spite of all the efforts, this is considered as a very difficult problem in coding theory and is still an open problem for most of the linear codes [11], [14], [25]. Furthermore, if t denotes the number of integers j satisfying and , then the code is called a t-weight code. Codes with a smaller value of t are called few weight codes. In particular, codes with are called equidistant or constant weight codes, which are useful in constructing combinatorial designs [15], [33] and generating goodsets of frequency hopping lists in radio networks [32]. The support of a word , denoted by , is defined as the set of its non-zero coordinate positions. Further, if are such that , then we say that the word u covers the word v. A codeword is said to be minimal if it only covers the codewords for all and it does not cover any other codeword of . The linear code is said to be minimal if every codeword of is minimal. Minimal linear codes are useful in designing secret sharing schemes with nice access structures [7], [26], [35] and in secure two-party computation [2], [9], and can be effectively decoded with the minimum distance decoding algorithm [1]. The problem of finding minimal linear codes over finite fields has recently attracted a lot of attention [1], [2], [8], [9], [12], [17], [25], [26], [27].
Now let , and let be positive integers coprime to q. Then a linear code of length over is called a Λ-multi-twisted (MT) code with block lengths if it is invariant under the Λ-multi-twisted shift operator on . In particular, when and , Λ-MT codes are called -double cyclic codes with block lengths [6], [29]. Multi-twisted (MT) codes over finite fields are introduced and studied by Aydin and Halilović [3]. These codes form an important class of linear codes and are generalizations of well-known classes of linear codes, such as constacyclic codes and generalized quasi-cyclic codes, having rich algebraic structures and containing record-breaker codes. In the same work, they also obtained subcodes of MT codes with best-known parameters over , over , over and optimal parameters over . Besides this, they proved that the code parameters over and over can not be attained by constacyclic or quasi-cyclic codes, which suggests that this larger class of MT codes is more promising to find codes with better parameters than the current best known linear codes. Later, Sharma et al. [30], [31] established algebraic structures of MT codes over finite fields and studied their dual codes with respect to the Euclidean and Hermitian inner products. They derived necessary and sufficient conditions for the existence of a Euclidean (resp. Hermitian) self-dual MT code, and obtained enumeration formulae for all Euclidean (resp. Hermitian) self-dual, self-orthogonal and LCD MT codes over finite fields. They also derived some sufficient conditions under which a MT code is Euclidean (resp. Hermitian) LCD. They determined the parity-check polynomial of each MT code and obtained a BCH type bound on their Hamming distances. They also determined generating sets of Euclidean (resp. Hermitian) dual codes of some MT codes from the generating sets of the corresponding MT codes. Besides this, they provided a trace description for all MT codes over finite fields by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. They also obtained a lower bound on their Hamming distances by viewing these codes as multilevel concatenated codes.
Next let , , and let be odd positive integers such that the irreducible factorization of the polynomial over is given by Further, let be a root of the polynomial , be a primitive element of , and let us write for some integer satisfying for each i. Patanker and Singh [29] recently determined Hamming weight distributions of -double cyclic codes (i.e., -MT codes over ) with block lengths under the assumption that there exists a least positive integer satisfying Here we assert that for . To prove this assertion, let be fixed. Now by (1.1), we observe that is the least positive integer satisfying , which implies that is a prime number. Since and , we note that is a primitive th root of unity. Without any loss of generality, we can assume that , i.e., we can take so that . Now one can easily see that conditions (1.1) and (1.2) hold for . Further, we see that does not satisfy the condition (1.1). Furthermore, for , we note that and the condition (1.2) implies that divides , which further implies that . From this, it follows that the condition (1.2) does not hold for any prime . This shows that conditions (1.1) and (1.2) are very heavy constraints and hold only when . In the light of this, Patanker and Singh [29] essentially determined Hamming weight distributions of some -double cyclic codes (i.e., -MT codes over ) of lengths only, which one can easily determine by direct computations and without applying deeper results on Gauss sums.
In another related direction, let us consider the cyclic trace code of length n over , defined as where for all and , where is a primitive element of , and is the trace function from onto for . By Delsarte's Theorem, one can see that the cyclic code has length and its parity check polynomial is given by , where and are minimal polynomials of and over , respectively. Li et al. [20] determined the Hamming weight distribution of the cyclic code when , , q is even, and . Vega [34] determined the Hamming weight distribution of the code when , and , and she also posed an open problem to determine the Hamming weight distribution of the code when , and . Later, Heng and Yue [16] determined the Hamming weight distribution of the code when , is any positive integer and . Recently, Li et al. [21] determined the Hamming weight distribution of the code in the case when and is a positive integer satisfying , and are not conjugates over and , provided either and or , and . Note that a Λ-MT code of length n and block lengths over is a cyclic code if and only if and . In particular, when , , and the Λ-MT code has exactly two non-zero constituents, say and corresponding to the irreducible factors and of the polynomial over respectively, we see, by Theorem 2.1, that , whose Hamming weight distribution (or equivalently, the Hamming weight enumerator) is known only in certain special cases [16], [20], [21], [34]. In this paper, we shall extend the technique employed in [16], [20], [21], [34] in order to relax the constraints imposed on and q in the cyclic case, and we shall explicitly determine Hamming weight enumerators (or equivalently, Hamming weight distributions) of several classes of MT codes over with at most two non-zero constituents, where each non-zero constituent has dimension 1.
The main goal of this paper is to explicitly determine Hamming weight enumerators of several classes of MT codes over finite fields with constituents of dimensions at most 1 and having at most two non-zero constituents. As applications, two classes of optimal equidistant linear codes and several other classes of minimal linear codes are identified within these classes of MT codes. The results obtained in [16], [20], [21], [34] follow from our results as special cases.
This paper is organized as follows: In Section 2, we state some preliminaries that are needed to derive our main results. In Section 3, we explicitly determine Hamming weights of non-zero codewords belonging to MT codes over finite fields with at most two non-zero constituents (see equation (3.2) and Theorem 3.1, Theorem 3.8). In Section 4, we apply the results derived in Section 3 and explicitly determine Hamming weight enumerators of several classes of MT codes having at most two non-zero constituents, where each non-zero constituent has dimension 1 (see Theorem 4.1, Theorem 4.8 and Remark 4.1). Among these classes of MT codes, we identify two classes of optimal equidistant linear codes that attain both the Griesmer and Plotkin bounds, and several other classes of minimal linear codes over finite fields that are useful in constructing secret sharing schemes with nice access structures (see Theorem 4.1, Theorem 4.5, Theorem 4.7, Theorem 4.8).
Section snippets
Some preliminaries
In this section, we shall state some preliminaries that are needed to obtain our main results. To begin with, we assume, throughout this paper, that is the finite field of order , where p is a prime and r is a positive integer. Let be positive integers coprime to q, and let . Let denote the vector space consisting of all n-tuples over . Let , where are non-zero elements of . Then a Λ-multi-twisted module V is an -module of the
Determination of the Hamming weights of codewords belonging to the MT codes with at most two non-zero constituents
In this section, we shall determine Hamming weights of non-zero codewords of several Λ-MT codes over having at most two non-zero constituents. To do this, without any loss of generality, we assume, throughout this paper, that the constituents of Λ-MT codes corresponding to the irreducible factors are zero. Then by Theorem 2.1, each Λ-MT code of length over is given by where
Hamming weight enumerators of MT codes with at most two non-constituents
In this section, we will explicitly determine Hamming weight enumerators of several classes of MT codes with at most two non-zero constituents, where each non-zero constituent has dimension 1. Among these classes of MT codes, we will further identify two classes of optimal equidistant linear codes that have nice connections with the theory of combinatorial designs and several other classes of minimal linear codes that are useful in constructing secret sharing schemes with nice access structures
Conclusion and future work
In this paper, Hamming weight enumerators of several MT codes over finite fields with at most two non-zero constituents and with each non-zero constituent of dimension 1 have been explicitly determined. As applications, several optimal equidistant linear codes and several minimal linear codes are identified within this class of MT codes. It is worth noting that the results obtained in [16], [20], [21], [34] follow from our results as special cases. It would be interesting to determine Hamming
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- 1
Research support by UGC, India, under the grant no. 2061641069 with reference no. 19/06/2016 (i) EU-V is gratefully acknowledged.
- 2
Research support by DST-SERB, India, under Grant no. MTR/2017/000358 is gratefully acknowledged.