Anticodes for the Grassman and bilinear forms graphs

Martin, William J.; Zhu, Xiaojun

doi:10.1007/bf01390772

Cited by 43 publications

(35 citation statements)

References 6 publications

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“…It arises the question if spread codes are perfect in this sense. The answer turns out to be negative in general and this result can be found in [MZ95].…”

Section: Non-perfectness Of a Spread Codementioning

confidence: 73%

Spread codes and spread decoding in network coding

Manganiello

Gorla

Rosenthal

2008

2008 IEEE International Symposium on Information Theory

122

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“…It arises the question if spread codes are perfect in this sense. The answer turns out to be negative in general and this result can be found in [MZ95].…”

Section: Non-perfectness Of a Spread Codementioning

confidence: 73%

Spread codes and spread decoding in network coding

Manganiello

Gorla

Rosenthal

2008

2008 IEEE International Symposium on Information Theory

122

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“…. , 8 ) specified in terms of the alternative parameters a, b, c 1 , c 2 , c 3 , c 4 of (12) and (13). It is easy to see…”

Section: Proof By Lemma 10 We Havementioning

confidence: 99%

Optimal tristance anticodes in certain graphs

Etzion

Schwartz

Vardy

2006

Journal of Combinatorial Theory, Series A

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For z 1 , z 2 , z 3 ∈ Z n , the tristance d 3 (z 1 , z 2 , z 3 ) is a generalization of the L 1 -distance on Z n to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode A d of diameter d is a subset of Z n with the property that d 3 (z 1 , z 2 , z 3 ) d for all z 1 , z 2 , z 3 ∈ A d . An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z 2 for all diameters d 1. We then generalize this result to two related distance models: a different distance structure on Z 2 where d(z 1 , z 2 ) = 1 if z 1 , z 2 are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z 2 is replaced by the hexagonal lattice A 2 . We also investigate optimal tristance anticodes in Z 3 and optimal quadristance anticodes in Z 2 , and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.

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“…Two such subspaces are adjacent, i.e., connected by an undirected edge, if and only if they intersect in a (k&1)-dimensional subspace. Martin and Zhu [6] proved that there are no nontrivial e-perfect codes in the Grassman graph. The size of optimal anticodes in the Grassman graph was determined by Frankl and Wilson [5] in their work on t-intersecting families.…”

Section: Introductionmentioning

confidence: 99%

Codes and Anticodes in the Grassman Graph

Schwartz

Etzion

2002

Journal of Combinatorial Theory, Series A

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Perfect codes and optimal anticodes in the Grassman graph G q (n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and``real'' Steiner systems. Elsevier Science

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Anticodes for the Grassman and bilinear forms graphs

Cited by 43 publications

References 6 publications

Spread codes and spread decoding in network coding

Spread codes and spread decoding in network coding

Optimal tristance anticodes in certain graphs

Codes and Anticodes in the Grassman Graph

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