## The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle

Publication ,  Journal Article
Calderbank, AR; Frankl, P; Graham, RL; Li, WCW; Shepp, LA
Published in: Journal of Algebraic Combinatorics: An International Journal
January 1, 1993

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, Körner, 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 bounds 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, (Formula presented.), q−1}n with the property that for every pair of distinct vectors x = (xi), y = (yi) (Formula presented.)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 (Formula presented.). © 1993, Kluwer Academic Publishers. All rights reserved.

## Published In

Journal of Algebraic Combinatorics: An International Journal

1572-9192

0925-9899

January 1, 1993

2

1

## Start / End Page

31 / 48

• General Mathematics
• 0101 Pure Mathematics

### Citation

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Calderbank, A. R., Frankl, P., Graham, R. L., Li, W. C. W., & Shepp, L. A. (1993). The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle. Journal of Algebraic Combinatorics: An International Journal, 2(1), 31–48. https://doi.org/10.1023/A:1022424630332
Calderbank, A. R., P. Frankl, R. L. Graham, W. C. W. Li, and L. A. Shepp. “The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle.” Journal of Algebraic Combinatorics: An International Journal 2, no. 1 (January 1, 1993): 31–48. https://doi.org/10.1023/A:1022424630332.
Calderbank AR, Frankl P, Graham RL, Li WCW, Shepp LA. The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle. Journal of Algebraic Combinatorics: An International Journal. 1993 Jan 1;2(1):31–48.
Calderbank, A. R., et al. “The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle.” Journal of Algebraic Combinatorics: An International Journal, vol. 2, no. 1, Jan. 1993, pp. 31–48. Scopus, doi:10.1023/A:1022424630332.
Calderbank AR, Frankl P, Graham RL, Li WCW, Shepp LA. The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle. Journal of Algebraic Combinatorics: An International Journal. 1993 Jan 1;2(1):31–48.

## Published In

Journal of Algebraic Combinatorics: An International Journal

1572-9192

0925-9899

January 1, 1993

2

1

31 / 48