On linear codes admitting large automorphism groups

Abstract

Linear codes with large automorphism groups are constructed. Most of them are suitable for permutation decoding. In some cases they are also optimal. For instance, we construct an optimal binary code of length (Formula presented.) and dimension (Formula presented.) having minimum distance (Formula presented.) and automorphism group isomorphic to (Formula presented.)

On the existence of PD-sets: Algorithms arising from automorphism groups of codes

“…On the other hand, the larger the automorphism group of a code, the more likely it is for a PD-set to exist; see [23,42,43,44]. Codes with large prescribed automorphism groups have been constructed in [51] by selecting suitable matrix group representations from the Atlas of Finite Groups Representations [1] and considering the corresponding transitive codes.…”

“…These problems have been studied by various authors for very particular classes of codes, see [7,15,16,17,19,25,27,29,30,31,32,33,34,35,42,43,44,53,57]. If the existence of a PD-set is known and the code has relatively small parameters, a computational approach to Problem (2) was proposed by the authors in [51]. This approach produced several examples of small PD-sets that, in some cases, can be proven to be the smallest possible.…”

“…This approach produced several examples of small PD-sets that, in some cases, can be proven to be the smallest possible. In general, it is convenient to start with a linear code with a large prescribed automorphism group and, not only because of permutation decoding, these codes have been widely investigated in the last years, see [2,4,6,9,8,10,11,12,13,14,18,21,24,36,37,38,39,40,41,49,50,51,52,54,56] and references therein. This paper deals with Problem (1).…”

$ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $

“…A survey of permutation decoding using codes that arise from combinatorial structures, such as designs, finite geometries and graphs, is given by Key in [6]. The problem of constructing a PD-set for a linear code admitting a large automorphism group has been addressed in [11,Section 10]. Furthermore, PD-sets for codes spanned by incidence matrices of incidence graphs of flag-transitive symmetric designs have been studied in [2].…”

Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

“…For more on the connection between arcs and secret sharing schemes, see, for instance, [9], [10], and [14]. For these reasons, in recent years there has been great interest in constructing new arcs in projective spaces; see [2], [7], [8], [11], [12], [13], [15]. Moreover, the automorphism group of a code can be used to decrease the computational complexity of encoding and decoding [17], so finding new examples of codes with large automorphism groups is useful in practice.…”