On the Nonperiodic Cyclic Equivalence Classes of Reed-Solomon Codes

H. Y. Song, I. S. Reed, S. W. Golomb

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21 Citations (Scopus)


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∑dn μ(d)q1+[k/d], where µ(d) is the Möbius 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-qk+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. Furthermore, for all possible values of q2, a number B(q) is determined such that |E“| = |V| for 1 ≤k ≤ B(q) and “||V| for kB(q).

Original languageEnglish
Pages (from-to)1431-1434
Number of pages4
JournalIEEE Transactions on Information Theory
Issue number4
Publication statusPublished - 1993 Jul

Bibliographical note

Funding Information:
Manuscript received April 9, 1992. This work was supported in part by the NSF under Grant NCR-9016340 and in part by the United States Office of Naval Research under Grant NOOO14-90-5-1341.

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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