Constructions of Formally Self-Dual Codes over Z4 and Their Weight Enumerators

Jinjoo Yoo; Yoonjin Lee; Boreum Kim

doi:10.1109/TIT.2017.2761388

Constructions of Formally Self-Dual Codes over Z₄ and Their Weight Enumerators

Jinjoo Yoo, Yoonjin Lee, Boreum Kim

Department of Mathematics

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

3 Scopus citations

Abstract

We present three explicit methods for construction of formally self-dual codes over Z₄. We characterize relations between Lee weight enumerators of formally self-dual codes of length n+Z4 and those of length n+2; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over Z₄. Using these three constructions, we can find free formally self-dual codes over Z₄, as well as non-free formally self-dual codes over Z₄ of all even lengths. We find free or non-free formally self-dual codes over Z₄ of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over Z₄ by using the Gray map.

Original language	English
Article number	8063420
Pages (from-to)	7667-7675
Number of pages	9
Journal	IEEE Transactions on Information Theory
Volume	63
Issue number	12
DOIs	https://doi.org/10.1109/TIT.2017.2761388
State	Published - Dec 2017

Bibliographical note

Funding Information:
Manuscript received January 20, 2016; revised July 7, 2017; accepted September 28, 2017. Date of publication October 9, 2017; date of current version November 20, 2017. Y. Lee was supported in part by the Basic Science Research Program through the National Research Foundation (NRF) of Korea, Ministry of Education, under Grant 2009-0093827, and in part by the National Research Foundation (NRF) of Korea, Korea government (MEST) under Grant NRF-2017R1A2B2004574.

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

Formally self-dual code
Gray map
Lee weight enumerator
code over Z
non-linear extremal binary formally self-dual code

Access to Document

10.1109/TIT.2017.2761388

Cite this

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title = "Constructions of Formally Self-Dual Codes over Z4 and Their Weight Enumerators",

abstract = "We present three explicit methods for construction of formally self-dual codes over Z4. We characterize relations between Lee weight enumerators of formally self-dual codes of length n+Z4 and those of length n+2; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over Z4. Using these three constructions, we can find free formally self-dual codes over Z4, as well as non-free formally self-dual codes over Z4 of all even lengths. We find free or non-free formally self-dual codes over Z4 of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over Z4 by using the Gray map.",

keywords = "Formally self-dual code, Gray map, Lee weight enumerator, code over Z, non-linear extremal binary formally self-dual code",

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

note = "Funding Information: Manuscript received January 20, 2016; revised July 7, 2017; accepted September 28, 2017. Date of publication October 9, 2017; date of current version November 20, 2017. Y. Lee was supported in part by the Basic Science Research Program through the National Research Foundation (NRF) of Korea, Ministry of Education, under Grant 2009-0093827, and in part by the National Research Foundation (NRF) of Korea, Korea government (MEST) under Grant NRF-2017R1A2B2004574. Publisher Copyright: {\textcopyright} 1963-2012 IEEE.",

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N2 - We present three explicit methods for construction of formally self-dual codes over Z4. We characterize relations between Lee weight enumerators of formally self-dual codes of length n+Z4 and those of length n+2; the first two construction methods are based on these relations. The last construction produces free formally self-dual codes over Z4. Using these three constructions, we can find free formally self-dual codes over Z4, as well as non-free formally self-dual codes over Z4 of all even lengths. We find free or non-free formally self-dual codes over Z4 of lengths up to ten using our constructions. In fact, we obtain 46 inequivalent formally self-dual codes whose minimum Lee weights are larger than self-dual codes of the same length. Furthermore, we find 19 non-linear extremal binary formally self-dual codes of lengths 12, 16, and 20, up to equivalence, from formally self-dual codes over Z4 by using the Gray map.

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