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Construction of relatively maximal, systematic codes of specified minimum distance from linear recurring sequences of maximal period

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Relative to a distance function which is both translation-invariant and expressible as the sum of the distances between coordinates, an upper bound is obtained for the size of certain(n, d)systematic codes. This bound is closely related to a result of M. Plotkin. It is shown that certain(n, d), systematic codes obtainable from linear recurring sequences are of maximal size in an appropriate class of systematic(n, d)codes when the distance function is translation-invariant and the sum of the corresponding coordinate distances. The results are specialized to the Hamming distance and to the cyclic distance of C. Y. Lee. Relative to the Hamming distance, the results are valid for an arbitrary Galois fieldGF(q). For the cyclic distance, however, the results are valid only for prime Galois fields and forGF(4). Moreover, it is shown that for the latter distance, it is impossible to set up a "translation-invariant, coordinate-sum" distance which is also cyclic for any nonprime Galois field exceptGF(4).

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Information Theory, IRE Transactions on  (Volume:6 ,  Issue: 5 )