Di stance-spectrum formulas on the largest minimum distance of block codes

Po-Ning Chen*, Tzong Yow Lee, Yunghsiang S. Han

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


A general formula for the asymptotic largest minimum distance (in block length) of deterministic block codes under generalized distance functions (not necessarily additive, symmetric, and bounded) is presented. As revealed in the formula, the largest minimum distance can be fully determined by the ultimate statistical characteristics of the normalized distance function evaluated under a properly chosen random-code generating distribution. Interestingly, the new formula has an analogous form to the general information-spectrum expressions of the channel capacity and the optimistic channel capacity, respectively derived by Verdu-Han [29] and Chen-Alajaji [7], [8]. As a result, a minor class of distance functions for which the largest minimum distance can be derived is characterized. A general Varshamov-Gilbert lower bound is next addressed. Some discussions on the tightness of the general Varshamov-Gilbert bound are also provided. Finally, lower bounds on the largest minimum distances for several specific block coding schemes are rederived in terms of the new formulas, followed by comparisons with the known results devoted to the same codes.

Original languageEnglish
Pages (from-to)869-885
Number of pages17
JournalIEEE Transactions on Information Theory
Issue number3
StatePublished - May 2000


  • Block codes
  • Information spectrum
  • Varshamov-Gilbert bound


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