TY - JOUR
T1 - Constructions of Formally Self-Dual Codes over Z4 and Their Weight Enumerators
AU - Yoo, Jinjoo
AU - Lee, Yoonjin
AU - Kim, Boreum
N1 - 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.
PY - 2017/12
Y1 - 2017/12
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.
AB - 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.
KW - Formally self-dual code
KW - Gray map
KW - Lee weight enumerator
KW - code over Z
KW - non-linear extremal binary formally self-dual code
UR - http://www.scopus.com/inward/record.url?scp=85031822051&partnerID=8YFLogxK
U2 - 10.1109/TIT.2017.2761388
DO - 10.1109/TIT.2017.2761388
M3 - Article
AN - SCOPUS:85031822051
SN - 0018-9448
VL - 63
SP - 7667
EP - 7675
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 12
M1 - 8063420
ER -