### 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 N_{q}(n) of a subset S_{n} of {0, 1, ..., q - 1}^{n} with the property that for every pair of distinct vectors x = (x_{i}), y = (y_{i}) member of S_{n}, we have x_{j} - y_{j} ≡ 1(mod q) for some j? For q = 3 (the cyclic triangle), we show N_{3}(n) ≅ 2^{n}. If however S_{n} is a subgroup, then we give a simple proof that |S_{n}| ≤ √3^{n}.

Original language | English (US) |
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Title of host publication | Proceedings of the 1993 IEEE International Symposium on Information Theory |

Publisher | Publ by IEEE |

Number of pages | 1 |

ISBN (Print) | 0780308786 |

Publication status | Published - Jan 1 1993 |

Event | Proceedings of the 1993 IEEE International Symposium on Information Theory - San Antonio, TX, USA Duration: Jan 17 1993 → Jan 22 1993 |

### Publication series

Name | Proceedings of the 1993 IEEE International Symposium on Information Theory |
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### Other

Other | Proceedings of the 1993 IEEE International Symposium on Information Theory |
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City | San Antonio, TX, USA |

Period | 1/17/93 → 1/22/93 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)

### Cite this

*Proceedings of the 1993 IEEE International Symposium on Information Theory*(Proceedings of the 1993 IEEE International Symposium on Information Theory). Publ by IEEE.