Loop transversal codes
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This dissertation discusses the performance of loop transversal codes (LT codes), linear error correcting block codes constructed with attention to the syndrome function rather than to the code itself. LT codes are compared to lexicodes. Binary lexicodes which are linear are shown to be identical to those LT codes which are constructed by a greedy syndrome construction algorithm. Proofs by Conway and Sloane, and Brualdi and Pless, that binary lexicodes and greedy codes in the white-noise case are linear are generalized to the binary non-white-noise case. Using this result, we prove that those binary LT codes which are constructed by the greedy syndrome construction algorithm for a given set of errors (white or non-white noise) are always identical to the lexicode designed to correct the same set of errors. The proof of this generalization uses a metric d[subscript]E which is a generalization of the Hamming metric for any set of errors E such that 0 ⊂ E and E is closed under negation. This metric has the property that, if E is the set of errors corrected by a code C, then decoding is identical to minimum distance decoding under the metric d[subscript]E.;Those LT codes which are constructed by the greedy algorithm are shown to be maximal among linear codes, and in the case that the set of errors is closed under scalar multiplication, LT codes so constructed are shown to be maximal among all codes, linear and non-linear.;Data for ternary LT codes are shown to compare well--in both white-noise and non-white-noise cases--to the best linear codes known and also to lexicodes, which in non-binary cases are not generally linear. The ternary LT code constructed by the greedy algorithm for random single and double errors produces the (perfect) ternary (11, 6, 5) Golay code, a (43, 34, 5) code (1 dimension better than previously known for n = 43 and d = 5), a (44, 35, 5) code, and a (45, 36, 5) code, each 2 dimensions better than any previously known.