New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2

In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.Comment: 10 pages, 5 table

Constructing formally self-dual codes from block ƛ-circulant matrices

In this work, construction methods for formally self-dual codes are generalized in the form of block lambda-circulant matrices. The constructions are applied over the rings F_2,R1 = F_2 + uF_2 and S = F_2[u]=(u^3-1). Using n-block lambda-circulant matrices for suitable integers n and units lambda, many binary FSD codes (as Gray images) with a higher minimum distance than best known self-dual codes of lengths 34, 40, 44, 54, 58, 70, 72 and 74 were obtained. In particular, ten new even FSD [72, 36, 14] codes were constructed together with eight new near-extremal FSD even codes of length 44 and twentyfive new near-extremal FSD even codes of length 36

New extremal binary self-dual codes of length 68

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F_2 + uF_2 + u^2F_2, where u^3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with beta=117, 120, 133 in W68;1 and with gamma = 1, beta=49, 57, 59 and codes with gamma=2, beta=69, 81 in W68;2 are constructed for the ?first time in the literature. In addition to these, some known such codes are reconstructed via this extension. The results are tabulated

New binary self-dual codes via a generalization of the four circulant construction

In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets $\mathbb{F}_2$, $\mathbb{F}_2+u\mathbb{F}_2$, $\mathbb{F}_4+u\mathbb{F}_4$, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.https://www.mathos.unios.hr/mc/index.php/mc/article/view/352

2^n Bordered Constructions of Self-Dual codes from Group Rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes length 64 and 68, constructing 30 new extremal self-dual codes of length 68