A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔc constructed from simplicial complexes in Fn2, where Δ is a simplicial complex in Fn2 and Δc the complement of Δ. We first find an explicit computable criterion for CΔ c to be optimal; this criterion is given in terms of the 2-adic valuation of Σi=1 s 2|Ai|-1, where the Ai 's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔc is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when F has exactly two maximal elements, we explicitly determine the weight distribution of CΔc. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.
Bibliographical noteFunding Information:
Manuscript received July 16, 2019; revised April 4, 2020; accepted April 24, 2020. Date of publication May 7, 2020; date of current version October 21, 2020. The work of Jong Yoon Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1D1A1B05030707. The work of Jungyun Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) under Grant NRF-2017R1A6A3A11030486. The work of Yoonjin Lee was supported in part by the National Research Foundation of Korea (NRF) through the Basic Science Research Program funded by the Ministry of Education under Grant 2019R1A6A1A11051177 and in part by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) under Grant NRF-2017R1A2B2004574. (Corresponding author: Jong Yoon Hyun.) Jong Yoon Hyun is with Konkuk University, Glocal Campus, Chungju 27478, South Korea (e-mail: firstname.lastname@example.org).
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- Griesmer code
- Optimal linear code
- simplicial complex
- weight distribution 2010 AMS Subject Classification 94B05