New Singly and Doubly Even Binary [72,36,12] Self-Dual Codes from M_2(R)G – Group Matrix Rings
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In this work, we present a number of generator matrices of the form [I_2n | τ_k(v)], where I_kn is the kn × kn identity matrix, v is an element in the group matrix ring M_2(R)G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [72,36,12] self-dual codes directly over the finite field 𝔽_2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.
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