Sequences of linear codes where the rate times distance grows rapidly

Faezeh Alizadeh; S. P. Glasby; Cheryl E. Praeger

Sequences of linear codes where the rate times distance grows rapidly

Faezeh Alizadeh, S. P. Glasby, Cheryl E. Praeger

School of Physics, Mathematics and Computing

Research output: Contribution to journal › Article › peer-review

Abstract

For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes (formula presented) with parameters (formula presented) grows quickly in the sense that (formula presented). Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F₂ and k_id_i/n_i is asymptotic to (formula presented).

Original language	English
Pages (from-to)	105-113
Number of pages	9
Journal	Journal of Algebra Combinatorics Discrete Structures and Applications
Volume	10
Issue number	2
Publication status	Published - 10 May 2023

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title = "Sequences of linear codes where the rate times distance grows rapidly",

abstract = "For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes (formula presented) with parameters (formula presented) grows quickly in the sense that (formula presented). Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F2 and kidi/ni is asymptotic to (formula presented).",

keywords = "Linear code, Parameters, Rate, Reed Muller",

author = "Faezeh Alizadeh and Glasby, {S. P.} and Praeger, {Cheryl E.}",

note = "Funding Information: Acknowledgement: The authors thank Patrick Sol{\'e} for his remarks and for alerting us to [10]. The first author is grateful to UWA and the Centre for the Mathematics of Symmetry and Computation, and also to Shahid Rajaee Teacher Training University for their financial support when she was a visiting scholar. The second and third authors acknowledge support from the Australian Research Council Discovery Project DP190100450. Publisher Copyright: {\textcopyright} 2023, Jacodesmath Institute. All rights reserved.",

year = "2023",

month = may,

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language = "English",

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journal = "Journal of Algebra Combinatorics Discrete Structures and Applications",

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T1 - Sequences of linear codes where the rate times distance grows rapidly

AU - Alizadeh, Faezeh

AU - Glasby, S. P.

AU - Praeger, Cheryl E.

N1 - Funding Information: Acknowledgement: The authors thank Patrick Solé for his remarks and for alerting us to [10]. The first author is grateful to UWA and the Centre for the Mathematics of Symmetry and Computation, and also to Shahid Rajaee Teacher Training University for their financial support when she was a visiting scholar. The second and third authors acknowledge support from the Australian Research Council Discovery Project DP190100450. Publisher Copyright: © 2023, Jacodesmath Institute. All rights reserved.

PY - 2023/5/10

Y1 - 2023/5/10

N2 - For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes (formula presented) with parameters (formula presented) grows quickly in the sense that (formula presented). Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F2 and kidi/ni is asymptotic to (formula presented).

AB - For a linear code C of length n with dimension k and minimum distance d, it is desirable that the quantity kd/n is large. Given an arbitrary field F, we introduce a novel, but elementary, construction that produces a recursively defined sequence of F-linear codes (formula presented) with parameters (formula presented) grows quickly in the sense that (formula presented). Another example of quick growth comes from a certain subsequence of Reed-Muller codes. Here the field is F = F2 and kidi/ni is asymptotic to (formula presented).

KW - Linear code

KW - Parameters

KW - Rate

KW - Reed Muller

UR - http://www.scopus.com/inward/record.url?scp=85160223576&partnerID=8YFLogxK

UR - https://arxiv.org/abs/2110.01277

M3 - Article

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VL - 10

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JO - Journal of Algebra Combinatorics Discrete Structures and Applications

JF - Journal of Algebra Combinatorics Discrete Structures and Applications

IS - 2

ER -

Sequences of linear codes where the rate times distance grows rapidly

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Complexity of group algorithms and statistical fingerprints of groups

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Sequences of linear codes where the rate times distance grows rapidly

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Complexity of group algorithms and statistical fingerprints of groups

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