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 C
_Δc
constructed from simplicial complexes in F
₂
ⁿ
, where Δ is a simplicial complex in F
₂
ⁿ
and Δ
^c
the complement of Δ. We first find an explicit computable criterion for C
_Δc
to be optimal; this criterion is given in terms of the 2-adic valuation of Σ
^s
_j=1
2
^|Ai|-1
, where the At'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
_Δc
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 C
_Δc
.We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.