Self-orthogonal codes over Z4 arising from the chain ring Z4[u]/〈u2+1〉

Boran Kim; Nayoung Han; Yoonjin Lee

doi:10.1016/j.ffa.2021.101972

Self-orthogonal codes over Z₄ arising from the chain ring Z₄[u]/〈u²+1〉

Boran Kim, Nayoung Han, Yoonjin Lee

Department of Mathematics

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

1 Scopus citations

Abstract

We find a building-up type construction method for self-orthogonal codes over Z₄ arising from the chain ring Z₄[u]/〈u²+1〉. Our construction produces self-orthogonal codes over Z₄ with increased lengths and free ranks from given self-orthogonal codes over Z₄ with smaller lengths and free ranks; in the most of the cases their minimum weights are also increased. Furthermore, any self-orthogonal code over Z₄ with generator matrix subject to certain conditions can be obtained from our construction. Employing our construction method, we obtain at least 125 new self-orthogonal codes over Z₄ up to equivalence; among them, there are 35 self-orthogonal codes which are distance-optimal. Furthermore, we have eight self-orthogonal codes, which are distance-optimal among all linear codes over Z₄ with the same type. As a method, we use additive codes over the finite ring Z₄[u]/〈u²+1〉 with generator matrices G satisfying GG^T=O, and we use a new Gray map from Z₄[u]/〈u²+1〉 to Z₄³ as well.

Original language	English
Article number	101972
Journal	Finite Fields and their Applications
Volume	78
DOIs	https://doi.org/10.1016/j.ffa.2021.101972
State	Published - Feb 2022

Bibliographical note

Funding Information:
Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574). Boran Kim is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A1A01060467) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(NRF-2021R1C1C2012517). Nayoung Han is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A3A13065516).

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Additive code
Chain ring
Gray map
Optimal code
Self-orthogonal code

Access to Document

10.1016/j.ffa.2021.101972

Cite this

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title = "Self-orthogonal codes over Z4 arising from the chain ring Z4[u]/〈u2+1〉",

abstract = "We find a building-up type construction method for self-orthogonal codes over Z4 arising from the chain ring Z4[u]/〈u2+1〉. Our construction produces self-orthogonal codes over Z4 with increased lengths and free ranks from given self-orthogonal codes over Z4 with smaller lengths and free ranks; in the most of the cases their minimum weights are also increased. Furthermore, any self-orthogonal code over Z4 with generator matrix subject to certain conditions can be obtained from our construction. Employing our construction method, we obtain at least 125 new self-orthogonal codes over Z4 up to equivalence; among them, there are 35 self-orthogonal codes which are distance-optimal. Furthermore, we have eight self-orthogonal codes, which are distance-optimal among all linear codes over Z4 with the same type. As a method, we use additive codes over the finite ring Z4[u]/〈u2+1〉 with generator matrices G satisfying GGT=O, and we use a new Gray map from Z4[u]/〈u2+1〉 to Z43 as well.",

keywords = "Additive code, Chain ring, Gray map, Optimal code, Self-orthogonal code",

author = "Boran Kim and Nayoung Han and Yoonjin Lee",

note = "Funding Information: Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574). Boran Kim is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A1A01060467) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(NRF-2021R1C1C2012517). Nayoung Han is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A3A13065516). Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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AU - Kim, Boran

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N1 - Funding Information: Yoonjin Lee is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(NRF-2017R1A2B2004574). Boran Kim is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A1A01060467) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)(NRF-2021R1C1C2012517). Nayoung Han is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1A6A3A13065516). Publisher Copyright: © 2021 Elsevier Inc.

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N2 - We find a building-up type construction method for self-orthogonal codes over Z4 arising from the chain ring Z4[u]/〈u2+1〉. Our construction produces self-orthogonal codes over Z4 with increased lengths and free ranks from given self-orthogonal codes over Z4 with smaller lengths and free ranks; in the most of the cases their minimum weights are also increased. Furthermore, any self-orthogonal code over Z4 with generator matrix subject to certain conditions can be obtained from our construction. Employing our construction method, we obtain at least 125 new self-orthogonal codes over Z4 up to equivalence; among them, there are 35 self-orthogonal codes which are distance-optimal. Furthermore, we have eight self-orthogonal codes, which are distance-optimal among all linear codes over Z4 with the same type. As a method, we use additive codes over the finite ring Z4[u]/〈u2+1〉 with generator matrices G satisfying GGT=O, and we use a new Gray map from Z4[u]/〈u2+1〉 to Z43 as well.

AB - We find a building-up type construction method for self-orthogonal codes over Z4 arising from the chain ring Z4[u]/〈u2+1〉. Our construction produces self-orthogonal codes over Z4 with increased lengths and free ranks from given self-orthogonal codes over Z4 with smaller lengths and free ranks; in the most of the cases their minimum weights are also increased. Furthermore, any self-orthogonal code over Z4 with generator matrix subject to certain conditions can be obtained from our construction. Employing our construction method, we obtain at least 125 new self-orthogonal codes over Z4 up to equivalence; among them, there are 35 self-orthogonal codes which are distance-optimal. Furthermore, we have eight self-orthogonal codes, which are distance-optimal among all linear codes over Z4 with the same type. As a method, we use additive codes over the finite ring Z4[u]/〈u2+1〉 with generator matrices G satisfying GGT=O, and we use a new Gray map from Z4[u]/〈u2+1〉 to Z43 as well.

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