Maximum weight spectrum codes

Abstract

A linear [n, k]q code C is said to be a full weight spectrum (FWS) code if there exist codewords of each weight less than or equal to n. In this brief communication we determine necessary and sufficient conditions for the existence of linear [n, k]q full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in P G(k − 1, q).Recently, Shi et. al. [10,9] studied a combinatorial problem concerning the maximum number L(k, q) of distinct weights a linear code of dimension k over GF (q) may realize. Obviously L(k, q) ≤ q k −1 q−1 . In [10], this bound is shown to be sharp for binary codes, and for all q-ary codes of dimension k = 2. We note that in 2015, Haily and Harzalla [7] also established the existence of binary codes meeting this bound. Shi et. al. went on to conjecture that the bound is sharp for all q and k. This conjecture was proved correct in [1]. Codes meeting this bound are called maximum weight spectrum (MWS) codes.A further refinement was also investigated in [10] by the introduction of the function L(n, k, q), denoting the maximum number of non-zero weights an [n, k] q code may have. They observed that an immediate upper bound is L(n, k, q) ≤ n. In this short communication we establish necessary and sufficient conditions for the existence of codes meeting this bound. Such codes will be said to be full weight spectrum (FWS) codes. MSC(2010): Primary: 94B05 ; Secondary: 94B65 and 94B25.

How Many Weights Can a Quasi-Cyclic Code Have?

“…They introduced the functions L(n, k, q) and L(k, q) which represent respectively the maximum number of nonzero weights of an [n, k] code, and the maximum number of nonzero weights of a k-dimensional code, with no restriction on the length. In that paper, some results on these two functions were provided, leaving as on open conjecture that the value L(k, q) is always equal to q k −1 q−1 , Such a conjecture was proved to be true in [3], where Alderson and Neri introduced the notion of maximum weight spectrum codes (or MWS codes in short). These codes were studied also by other authors for their combinatorial interest (see [24], [9], [1]).…”

“…In that paper, some results on these two functions were provided, leaving as on open conjecture that the value L(k, q) is always equal to q k −1 q−1 , Such a conjecture was proved to be true in [3], where Alderson and Neri introduced the notion of maximum weight spectrum codes (or MWS codes in short). These codes were studied also by other authors for their combinatorial interest (see [24], [9], [1]). Moreover, the function L(n, k, q) was also studied in [28], and new answers were then provided by Alderson in [1].…”

“…These codes were studied also by other authors for their combinatorial interest (see [24], [9], [1]). Moreover, the function L(n, k, q) was also studied in [28], and new answers were then provided by Alderson in [1]. He defined and studied the combinatorial properties of full weight spectrum codes (or FWS codes in short), which are codes of length n with exactly n nonzero weights.…”

Codes with few weights arising from linear sets

“…see [2]. Note that the cardinalities above are counted with multiplicities in the case of a multiset.…”

On the Weights of General MDS Codes

“…Given a set of positive integers, S, is it possible to construct a code whose set of non-zero weights is S? More recent works, such as [2], [3], [4], and [5] investigate upper bounds on the size of the weight spectra of linear codes, and the existence of maximum weight spectrum (MWS) codes, and full weight spectrum (FWS) codes. Shi et.…”