An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Bohyun Kim; Yoonjin Lee; Jinjoo Yoo

doi:10.1016/j.ffa.2021.101923

An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Bohyun Kim, Yoonjin Lee, Jinjoo Yoo

Department of Mathematics

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

2 Scopus citations

Abstract

Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field F_p^m. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring R_p,m=F_p^m[u,v]/〈u²=v²=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of R_p,m, a,b,c,d∈F_p^m, and a is nonzero. We then find a Gray map from R_p,m[x]/〈xⁿ−α〉 (with respect to homogeneous weights) to F_p^m[x]/〈x^{p^3m+1n}−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over R_p,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over F_p^m of length p^3m+1 and index p^3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.

Original language	English
Article number	101923
Journal	Finite Fields and their Applications
Volume	76
DOIs	https://doi.org/10.1016/j.ffa.2021.101923
State	Published - Dec 2021

Bibliographical note

Funding Information:
Y. Lee is a corresponding author and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) ( NRF-2017R1A2B2004574 ) and also by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177 ). J. Yoo is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ) and also by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2021R1I1A1A01047765 ).

Publisher Copyright:
© 2021

Keywords

Gray map
Griesmer code
Quasi-cyclic code
Self-orthogonal code

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10.1016/j.ffa.2021.101923

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title = "An infinite family of Griesmer quasi-cyclic self-orthogonal codes",

abstract = "Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉 (with respect to homogeneous weights) to Fpm[x]/〈xp3m+1n−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over Rp,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over Fpm of length p3m+1 and index p3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.",

keywords = "Gray map, Griesmer code, Quasi-cyclic code, Self-orthogonal code",

author = "Bohyun Kim and Yoonjin Lee and Jinjoo Yoo",

note = "Funding Information: Y. Lee is a corresponding author and supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) ( NRF-2017R1A2B2004574 ) and also by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177 ). J. Yoo is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1016649 ) and also by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2021R1I1A1A01047765 ). Publisher Copyright: {\textcopyright} 2021",

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N2 - Our aim for this paper is to find the construction method for quasi-cyclic self-orthogonal codes over the finite field Fpm. We first explicitly determine the generators of α-constacyclic codes over the finite Frobenius non-chain ring Rp,m=Fpm[u,v]/〈u2=v2=0,uv=vu〉, where m is a positive integer, α=a+ub+vc+uvd is a unit of Rp,m, a,b,c,d∈Fpm, and a is nonzero. We then find a Gray map from Rp,m[x]/〈xn−α〉 (with respect to homogeneous weights) to Fpm[x]/〈xp3m+1n−a〉 (with respect to Hamming weights), which is linear and preserves minimum weights. We present an efficient algorithm for finding the Gray images of α-constacyclic codes over Rp,m of length n, which produces infinitely many quasi-cyclic self-orthogonal codes over Fpm of length p3m+1 and index p3m. In particular, some family turns out to be “Griesmer” codes; these Griesmer quasi-cyclic self-orthogonal codes are “new” codes compared with previously known Griesmer codes of dimension 4.

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An infinite family of Griesmer quasi-cyclic self-orthogonal codes

Abstract

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Keywords

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