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