Infinite Families of Optimal Linear Codes Constructed From Simplicial Complexes

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&g...

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Veröffentlicht in:IEEE transactions on information theory Jg. 66; H. 11; S. 6762 - 6773
Hauptverfasser: Hyun, Jong Yoon, Lee, Jungyun, Lee, Yoonjin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.11.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
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
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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2020.2993179