Infinite Families of Optimal Linear Codes Constructed From Simplicial Complexes - Hyun, Jong Yoon; Lee, Jungyun; Lee, Yoonjin (IEEE transactions on information theory, 2020)
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 C_{Δ ^{c}} constructed from simplicial complexes in \mathbb {F}^{n}_{2} , where Δ is a simplicial complex in \mathbb {F}^{n}_{2} and Δ ^{c} the complement of Δ . We first find an explicit computable criterion for C_{Δ ^{c}} to be optimal; this criterion is given in terms of the 2-adic valuation of \sum _{i=1}^{s} 2^{|A_{i}|-1} , where the A_{i} '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 C_{Δ ^{c}} is a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when \mathcal {F} has exactly two maximal elements, we explicitly determine the weight distribution of C_{Δ ^{c}} . We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.