Sperner capacity of linear and nonlinear codes for the cyclic triangle

A. R. Calderbank, R. L. Graham, L. A. Shepp, P. Frank, W. C.W. Li

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Shannon introduced the concept of zero-error capacity of a discrete memoryless channel. The channel determines an undirected graph on the symbol alphabet, where adjacency means that symbols cannot be confused at the receiver. The zero-error or Shannon capacity is an invariant of this graph. Gargano, Koerner, and Vaccaro have recently extended the concept of Shannon capacity to directed graphs. Their generalization of Shannon capacity is called Sperner capacity. We resolve a problem posed by these authors by giving the first example (the two orientations of the triangle) of a graph where the Sperner capacity depends on the orientations of the edges. Sperner capacity seems to be achieved by nonlinear codes, whereas Shannon capacity seems to be attainable by linear codes. In particular, linear codes do not achieve Sperner capacity for the cyclic triangle. We use Fourier analysis or linear programming to obtain the best upper bounds for linear codes. The bound for unrestricted codes are obtained from rank arguments, eigenvalue interlacing inequalities and polynomial algebra. The statement of the cyclic q-gon problem is very simple: what is the maximum size Nq(n) of a subset Sn of {0, 1, ..., q - 1}n with the property that for every pair of distinct vectors x = (xi), y = (yi) member of Sn, we have xj - yj ≡ 1(mod q) for some j? For q = 3 (the cyclic triangle), we show N3(n) ≅ 2n. If however Sn is a subgroup, then we give a simple proof that |Sn| ≤ √3n.

Original languageEnglish (US)
Title of host publicationProceedings of the 1993 IEEE International Symposium on Information Theory
PublisherPubl by IEEE
Number of pages1
ISBN (Print)0780308786
Publication statusPublished - Jan 1 1993
EventProceedings of the 1993 IEEE International Symposium on Information Theory - San Antonio, TX, USA
Duration: Jan 17 1993Jan 22 1993

Publication series

NameProceedings of the 1993 IEEE International Symposium on Information Theory

Other

OtherProceedings of the 1993 IEEE International Symposium on Information Theory
CitySan Antonio, TX, USA
Period1/17/931/22/93

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All Science Journal Classification (ASJC) codes

  • Engineering(all)

Cite this

Calderbank, A. R., Graham, R. L., Shepp, L. A., Frank, P., & Li, W. C. W. (1993). Sperner capacity of linear and nonlinear codes for the cyclic triangle. In Proceedings of the 1993 IEEE International Symposium on Information Theory (Proceedings of the 1993 IEEE International Symposium on Information Theory). Publ by IEEE.