An Assmus–Mattson theorem for codes over commutative association schemes

Morales, John Vincent S.; Tanaka, Hajime

doi:10.1007/s10623-017-0376-y

Cited by 4 publications

(3 citation statements)

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“…This theorem gives a criterion for identifying a t-design as a collection of supports of codewords of fixed weight in a linear code. Since its publication in 1969 it has seen a number of generalisations [6,13] and has been used widely to obtain new constructions of designs [8,16]. In one of these results [3], the authors define a weighted t-design as a generalisation of a classical t-design and give criteria for identifying such an object among the dependent sets of a matroid of a fixed cardinality.…”

Section: Introductionmentioning

confidence: 99%

Weighted Subspace Designs from $q$-Polymatroids

Byrne,

Ceria,

Ionica

et al. 2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Weighted Subspace Designs from $q$-Polymatroids

Byrne,

Ceria,

Ionica

et al. 2021

Preprint

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“…Another coding theory application of the symmetry reduction technique are new approaches to the Assmus-Mattson Theorem by Tanaka [63] and by Morales and Tanaka [49]…”

Section: Extensions and Ramificationsmentioning

confidence: 99%

Semidefinite programming bounds for error-correcting codes

Vallentin

2019

Preprint

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This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Solé. This book will collect short but foundational articles, emphasizing definitions, examples, exhaustive references, and basic facts on the model of the Handbook of Finite Fields. The target audience of the Encyclopedia is upper level undergraduates and graduate students.

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“…The celebrated Assmus-Mattson Theorem [2] is among the best-known results in the theory of codes and designs. It has led to several constructions of t-designs [2,3,14,21,26] and has been the subject of a number of generalizations [4,15,16,27,34].…”

Section: Introductionmentioning

confidence: 99%

An Assmus--Mattson Theorem for Rank Metric Codes

Byrne¹,

Ravagnani²

2019

SIAM J. Discrete Math.

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A t-(n, d, λ) design over Fq, or a subspace design, is a collection of ddimensional subspaces of F n q , called blocks, with the property that every t-dimensional subspace of F n q is contained in the same number λ of blocks. A collection of matrices in F n×m q is said to hold a t-design over Fq if the set of column spaces of its elements forms the blocks of a subspace design. We use notions of puncturing and shortening of rank metric codes and the rank-metric MacWilliams identities to establish conditions under which the words of a given rank in a linear rank metric code hold a t-design over Fq. We show that for Fqm -linear vector rank metric codes, the property of a code being MRD is equivalent to its minimal weight codewords holding trivial subspace designs and show that this characterization does not hold for Fq-linear matrix MRD code that are not linear over Fqm .2010 Mathematics Subject Classification. 11T71, 05B05, 05E20.

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An Assmus–Mattson theorem for codes over commutative association schemes

Cited by 4 publications

References 36 publications

Weighted Subspace Designs from $q$-Polymatroids

Weighted Subspace Designs from $q$-Polymatroids

Semidefinite programming bounds for error-correcting codes

An Assmus--Mattson Theorem for Rank Metric Codes

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