Double bordered constructions of self-dual codes from group rings over Frobenius rings

Abstract

In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F 2 + uF 2 and F 4 + uF 4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables. Keywords Group rings • Self-dual codes • Codes over rings • Extremal codes • Bordered constructions Mathematics Subject Classification (2010) 94B05 • 94B15 This research was supported by the London Mathematical Society (International Short Visits-Scheme 5).

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

“…In this section, we apply the theorems given in the previous section to obtain many new best known binary self-dual codes. In particular, we obtain 28 singly-even [80,40,14] codes, 107 [84, 42, 14] codes, 105 singly-even [96,48,16] codes and 121 doubly-even [96, 48, 16] codes. We search for these codes using MATLAB and determine their properties using Qextension [3] and Magma [2].…”

“…where α, β ∈ Z. Previously known (α, β) values for weight enumerator W80 can be found online at [29] (see [19,31,13,16,15,28,17]). We obtain 28 new best known singly-even binary self-dual codes of length 80 which have weight enumerator W80 for Of the 28 new codes, 19 are constructed by applying Theorem 3.2 over F4 (Table 2); 4 are constructed by applying Theorem 3.4 over F2 + uF2 (Table 3) and 5 are constructed by applying Theorem 3.4 over F4 (Table 4).…”

“…Using generator matrices of the form (In | Ω(v)) for a number of different composite matrices Ω(v), we find many self-dual codes with weight enumerator parameters of previously unknown values (relative to referenced sources). In total, 361 new codes are found, including 28 singly-even binary self-dual [80,40,14] codes, 107 binary self-dual [84,42,14] codes, 105 singly-even binary self-dual [96,48,16] codes and 121 doublyeven binary self-dual [96,48,16] codes.…”

New binary self-dual codes of lengths 80, 84 and 96 from composite matrices

“…where α, β ∈ Z. Previously known (α, β) values for weight enumerator W 80 can be found online at [30] (see [14,[16][17][18]20,29,33]).…”

Binary self-dual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours

“…A substantial amount of work has been done on constructing self-dual codes having an automorphism of odd prime order [36,62,63,[66][67][68]. Recently, a strong connection between group rings and self-dual codes was established [21] which has been utilised to develop a number of different techniques for constructing extremal binary self-dual codes [14,25,30,32].…”