On the minimum number of minimal codewords

Cruz, Romar dela; Kiermaier, Michael; Kurz, Sascha; Wassermann, Alfred

doi:10.3934/amc.2020130

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…The latter shows that a linear code in which the minimum and maximum weight are close enough to each other is necessarily minimal. Recently, estimates for the number of minimal codewords in a given code have been also found; see [18].…”

Section: Introductionmentioning

confidence: 99%

Three Combinatorial Perspectives on Minimal Codes

Alfarano¹,

Borello²,

Neri³

et al. 2020

Preprint

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We develop three approaches of combinatorial flavour to study the structure of minimal codes, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-Füredi Theorem and the Combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. Applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.

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Section: Introductionmentioning

confidence: 99%

Three Combinatorial Perspectives on Minimal Codes

Alfarano¹,

Borello²,

Neri³

et al. 2020

Preprint

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show abstract

“…The latter shows that a linear code in which the minimum and the maximum weight are close enough is necessarily minimal. Recently, estimates for the number of minimal codewords in a given code have been also found; see [21].…”

mentioning

confidence: 99%

Three Combinatorial Perspectives on Minimal Codes

Alfarano¹,

Borello²,

Neri³

et al. 2022

SIAM J. Discrete Math.

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“…More generally, let M q (n, k) be the maximum and m q (n, k) the minimum of M (C) for all [n, k] q codes C. Bounds and some exact values on M q (n, k) and m q (n, k) can be found in [2,4,5,6,11]. Obviously, we have…”

Section: Preliminariesmentioning

confidence: 99%

On the number of minimal codewords in codes generated by the adjacency matrix of a graph

Kurz

2022

Discrete Applied Mathematics

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On the minimum number of minimal codewords

Cited by 5 publications

References 20 publications

Three Combinatorial Perspectives on Minimal Codes

Three Combinatorial Perspectives on Minimal Codes

Three Combinatorial Perspectives on Minimal Codes

On the number of minimal codewords in codes generated by the adjacency matrix of a graph

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