Infinite Families of Optimal Linear Codes Constructed From Simplicial Complexes

• Published in: IEEE transactions on information theory Vol. 66; no. 11; pp. 6762 - 6773
• Main Authors: Hyun, Jong Yoon, Lee, Jungyun, Lee, Yoonjin
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 3G mobile communication; 94A60; Binary codes; Binary system; Codes; Cost accounting; Criteria; Cryptography; Generators; Griesmer code; Linear codes; Mathematics; Optimal linear code; simplicial complex; weight distribution 2010 AMS Subject Classification 94B05
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2020.2993179

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 <inline-formula> <tex-math notation="LaTeX">C_{\Delta ^{c}} </tex-math></inline-formula> constructed from simplicial complexes in <inline-formula> <tex-math notation="LaTeX">\mathbb {F}^{n}_{2} </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula> is a simplicial complex in <inline-formula> <tex-math notation="LaTeX">\mathbb {F}^{n}_{2} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">\Delta ^{c} </tex-math></inline-formula> the complement of <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>. We first find an explicit computable criterion for <inline-formula> <tex-math notation="LaTeX">C_{\Delta ^{c}} </tex-math></inline-formula> to be optimal; this criterion is given in terms of the 2-adic valuation of <inline-formula> <tex-math notation="LaTeX">\sum _{i=1}^{s} 2^{|A_{i}|-1} </tex-math></inline-formula>, where the <inline-formula> <tex-math notation="LaTeX">A_{i} </tex-math></inline-formula>'s are maximal elements of <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula>. In particular, we find that <inline-formula> <tex-math notation="LaTeX">C_{\Delta ^{c}} </tex-math></inline-formula> is a Griesmer code if and only if the maximal elements of <inline-formula> <tex-math notation="LaTeX">\Delta </tex-math></inline-formula> are pairwise disjoint and their sizes are all distinct. Specially, when <inline-formula> <tex-math notation="LaTeX">\mathcal {F} </tex-math></inline-formula> has exactly two maximal elements, we explicitly determine the weight distribution of <inline-formula> <tex-math notation="LaTeX">C_{\Delta ^{c}} </tex-math></inline-formula>. We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.