The nonexistence of an additive quaternary [15,5,9]-code

Abstract

Available online xxxx Communicated by James W.P. Hirschfeld MSC: 94B60 51E20We show that no additive [15,5,9] 4 -code exists. As a consequence the largest dimension k such that an additive quaternary [15, k, 9] 4 -code exists is k = 4.5.

Additive Quaternary Codes Related to Exceptional Linear Quaternary Codes

“…By Theorem 4 this is the dual of the elliptic quadric code. It suffices therefore to start from this set of 17 lines and to check by a simple computer program that there is no line in P G (7,2) which completes it to a [18, 4] 4 -code of strength 3.…”

“…In fact we work with the dual, a [22, 4.5] 4 -code of strength 3. We work in P G (8,2). The underlying vector space is V (9, 2) with basis e 1 , .…”

“…It states that a family of codelines in P G(2k − 1, 2) generates an F 4 -linear code if and only if each subset of codelines generates a vector space of even binary dimension. Among the exceptional linear quaternary codes are the hexacode [6,3,4] 4 (geometrically the hyperoval in P G (2,4), see [1], Section 17.1 and Exercise 3.7.4), the extended cyclic code [12,6,6] 4 (see [1], Section 13.4 and Exercise 17.1.10) and the dual pair of codes [17,4,12] 4 and [17, 13, 4] 4 described by the elliptic quadric in P G(3, 4) (see Section 17.3 of [1]). In the present paper we study the generalization from linear quaternary to additive codes of code parameters related to those of our exceptional linear codes.…”

“…Such additive codes are new and their parameters cannot be reached by linear codes. For the parameters of optimal additive codes of small dimension see [2] and the references therein. For simplicity we denote P G(i − 1, 2) also as V i .…”

On Construction of Good Quaternary Additive Codes

“…In [3][4][5][6][7], some authors discussed the parameters of optimal additive codes and explicit construction of short length additive codes with length 15 d n and leaves several cases unsolved. The results of [3][4][5] shows that, for length 15 d n , there are quaternary additive codes have double codewords than their corresponding optimal linear code of the same length and minimum distance.…”