On the Distance Distribution of Codes

Gil Kalai, Nathan Linial

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34 Scopus citations

Abstract

The distance distribution of a binary code C is the sequence (Bi)in=0 defined as follows: Let Bi(w) be the number of codewords at distance i from the codeword w, and let Bi be the average of Bi(w) over all w in C. In this correspondence we study the distance distribution for codes of length n and minimal distance δn, with δ > 0 fixed and n → ∞. Our main aim is to relate the size of the code with the distribution of distances near the minimal distance.

Original languageAmerican English
Pages (from-to)1467-1472
Number of pages6
JournalIEEE Transactions on Information Theory
Volume41
Issue number5
DOIs
StatePublished - Sep 1995

Bibliographical note

Funding Information:
Gilbert's proof of this bound is simply to "grow" a code, by always adding new codewords subject only to the constraint that no distances smaller than h It occur. Despite its extreme simplicity, Manuscript received August 29, 1993; revised Nov. 18, 1994. 1991 Mathe- matics Subject Classification. Primary 94B65, Secondary 05B40, 52C17. This work was supported in part by the Binational Israel-US Science Foundation and by Israeli Academy of Sciences and Humanities. G. Kalai is w8ith the Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel. N. Linial is with the Institute of Computer Science, Hebrew University. Jerusalem 9 1904, Israel. IEEE Log Number 9413357.

Keywords

  • Binary codes
  • distance distribution
  • linear programming bounds

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