Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes

Jong Yoon Hyun; Jungyun Lee; Yoonjin Lee

doi:10.1109/TIT.2020.2993179

Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes

Jong Yoon Hyun, Jungyun Lee, Yoonjin Lee

Department of Mathematics

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

16 Scopus citations

Abstract

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai 's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔc is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of CΔc. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

Original language	English
Article number	9088995
Pages (from-to)	6762-6773
Number of pages	12
Journal	IEEE Transactions on Information Theory
Volume	66
Issue number	11
DOIs	https://doi.org/10.1109/TIT.2020.2993179
State	Published - Nov 2020

Bibliographical note

Funding Information:
Manuscript received July 16, 2019; revised April 4, 2020; accepted April 24, 2020. Date of publication May 7, 2020; date of current version October 21, 2020. The work of Jong Yoon Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1D1A1B05030707. The work of Jungyun Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1A6A3A11030486. The work of Yoonjin Lee was supported in part by the National Research Foundation of Korea (NRF) through the Basic Science Research Program funded by the Ministry of Education under Grant 2019R1A6A1A11051177 and in part by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) under Grant NRF-2017R1A2B2004574. (Corresponding author: Jong Yoon Hyun.) Jong Yoon Hyun is with Konkuk University, Glocal Campus, Chungju 27478, South Korea (e-mail: hyun33@kku.ac.kr).

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

94A60
Griesmer code
Optimal linear code
simplicial complex
weight distribution 2010 AMS Subject Classification 94B05

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10.1109/TIT.2020.2993179

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title = "Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes",

abstract = "A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai 's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔc is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of CΔc. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes. ",

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author = "Hyun, {Jong Yoon} and Jungyun Lee and Yoonjin Lee",

note = "Funding Information: Manuscript received July 16, 2019; revised April 4, 2020; accepted April 24, 2020. Date of publication May 7, 2020; date of current version October 21, 2020. The work of Jong Yoon Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1D1A1B05030707. The work of Jungyun Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1A6A3A11030486. The work of Yoonjin Lee was supported in part by the National Research Foundation of Korea (NRF) through the Basic Science Research Program funded by the Ministry of Education under Grant 2019R1A6A1A11051177 and in part by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) under Grant NRF-2017R1A2B2004574. (Corresponding author: Jong Yoon Hyun.) Jong Yoon Hyun is with Konkuk University, Glocal Campus, Chungju 27478, South Korea (e-mail: hyun33@kku.ac.kr). Publisher Copyright: {\textcopyright} 1963-2012 IEEE.",

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N2 - A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai 's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔc is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of CΔc. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

AB - A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai 's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔc is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of CΔc. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

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M1 - 9088995

ER -

Infinite Families of Optimal Linear Codes Constructed from Simplicial Complexes

Abstract

Bibliographical note

Keywords

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