### Abstract

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)q^{1+[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| = (q^{k+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. 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 language | English |
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Pages (from-to) | 1431-1434 |

Number of pages | 4 |

Journal | IEEE Transactions on Information Theory |

Volume | 39 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1993 Jul |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*IEEE Transactions on Information Theory*,

*39*(4), 1431-1434. https://doi.org/10.1109/18.243465

}

*IEEE Transactions on Information Theory*, vol. 39, no. 4, pp. 1431-1434. https://doi.org/10.1109/18.243465

**On the Nonperiodic Cyclic Equivalence Classes of Reed-Solomon Codes.** / Song, H. Y.; Reed, I. S.; Golomb, S. W.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the Nonperiodic Cyclic Equivalence Classes of Reed-Solomon Codes

AU - Song, H. Y.

AU - Reed, I. S.

AU - Golomb, S. W.

PY - 1993/7

Y1 - 1993/7

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=0027799665&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027799665&partnerID=8YFLogxK

U2 - 10.1109/18.243465

DO - 10.1109/18.243465

M3 - Article

AN - SCOPUS:0027799665

VL - 39

SP - 1431

EP - 1434

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 4

ER -