Loop transversal codes over finite rings
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In 1992, J. D. H. Smith introduced the concept of loop transversals to linear codes. The set of errors is given the structure of a loop transversal to the linear code as a subgroup of the channel. A greedy algorithm for specifying the loop structure, and thus for the construction of loop transversal codes, was discussed in the successive papers by F.-L. Hsu, F. A. Hummer and J. D. H. Smith. In 1996 paper by Hummer and Smith, the focus was mainly on error correction, in the white noise case constructing codes with odd minimum distance. In this paper an algorithm to compute loop transversal codes with even minimum distance is given. To give a complete description of the greedy LT codes an algorithm to compute a generator matrix is presented here. A vast number of optimal and best known codes are produced using the greedy loop transversal algorithm. Along with these codes the extended Hamming codes, the binary Golay [24, 12, 8], Reed-Muller [16, 5, 8], the quadratic residue [18, 9, 6], and the ternary Golay [12, 6, 6] codes are obtained. Among the quaternary codes a code equivalent to the octacode is obtained, and consequently the Nordstrom-Robinson code as its binary image under the Gray map. Record breaking codes over the 7-ary alphabet are also obtained.