From random matrix theory to coding theory: Volume of a metric ball in unitary group

Lu Wei*, Renaud Alexandre Pitaval, Jukka Corander, Olav Tirkkonen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)


Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, new results for the volume of a metric ball in unitary group are derived via tools from random matrix theory. The first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. A simple but accurate limiting volume formula is then obtained by invoking Szeg's strong limit theorem for large Toeplitz matrices. The derived asymptotic volume formula enables analytical evaluation of some coding-theoretic bounds of unitary codes. In particular, the Gilbert-Varshamov lower bound and the Hamming upper bound on the cardinality as well as the resulting bounds on code rate and minimum distance are derived. Moreover, bounds on the scaling law of code rate are found. Finally, a closed-form bound on the diversity sum relevant to unitary space-time codes is obtained, which was only computed numerically in the literature.

Original languageEnglish
Article number7876735
Pages (from-to)2814-2821
Number of pages8
JournalIEEE Transactions on Information Theory
Issue number5
Publication statusPublished - 1 May 2017
MoE publication typeA1 Journal article-refereed


  • Coding-theoretic bounds
  • random matrix theory
  • unitary group
  • volume of metric balls


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