The maximum number of minimal codewords in an[n,k]-code

Abstract

Upper and lower bounds are derived for the quantity in the title, which is tabulated for modest values of n and k. An application to graphs with many cycles is given.2010 Mathematics Subject Classification. Primary 94A10; Secondary 05C38,05B35.

On the minimum number of minimal codewords

“…Hence, M q (n, k) ≤ n k−1 . This result was first proved in [14] for matroids, and an alternative proof was given in [4] for binary codes. Equality is attained when C is a maximum distance separable (MDS) code, that is, a code with minimum distance d = n − k + 1.…”

“…We have m q (k, k) = k since for C = F k q , the minimal codewords are the vectors of weight one. Similarly, we define M q (n, k) to be the maximum of M (C) for all projective [n, k] q codes C. This quantity was studied in [4,5] for the case of binary codes. The focus of this work is on m q (n, k) and it is interesting to note that finding the minimum of M (C) is one of the problems raised in [17], the paper that introduced the concept of minimal codewords.…”

“…Minimal codewords have gained interest for various reasons. They can be viewed as circuits in matroids and also as cycles in graphs (see for instance [3,4,5,10,14,15,18]). In coding theory, minimal codewords were first used in decoding algorithms [1,2,17].…”

“…The authors in [3,4,5] initiated a new research direction by studying the maximum and minimum number of minimal codewords in binary linear codes. Given the length and dimension, bounds and some exact values were presented.…”

Cycles in 5-connected triangulations

“…Determining the number of cycles in a graph is a natural problem to consider in Graph Theory and has considerable application in, for example, Coding Theory (see [1][2][3]). We E-mail addresses: adelnife2@yahoo.com (A. Alahmadi), raldred@maths.otago.ac.nz (R.E.L.…”