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On the nonperiodic cyclic equivalence classes of Reed-Solomon codes

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2 Author(s)
H. Y. Song ; Dept. of EE-Syst., Univ. of Southern California, Los Anglees, CA ; S. W. Golomb

Picking up exactly one member from each of the nonperiodic cyclic equivalence classes of an (n, k+1) Reed-Solomon code E over GF(q) gives a code, E", which has bounded Hamming correlation values and the self-synchronizing property. The exact size of E" is shown to be (1/n) Σd|n μ(d)q1+kd/, where μ(d) is the Mobius function, (x) is the integer part of x, and the summation is over all the divisors d of n=q-1. A construction for a subset V of E is given to prove that |E"|⩾|V|=(qk+1-q k+1-N)/(q-1) where N is the number of integers from 1 to k which are relatively prime to q-1. A necessary and sufficient condition for |E"|=| V| is proved and some special cases are presented with examples. For all possible values of q>2, a number B(q) is determined such that |E"|=|V| for 1 ⩽kB(q ) and |E"|>|V| for k>B( q)

Published in:

IEEE Transactions on Information Theory  (Volume:39 ,  Issue: 4 )