New binary quantum stabilizer codes from the binary extremal self-dual $$[48, 24, 12]$$ [ 48 , 24 , 12 ] code

Abstract

This paper is devoted to constructing binary quantum stabilizer codes based on the binary extremal self-dual code of parameters $$[48, 24, 12]$$[48,24,12] by Steane's construction. First, we provide an explicit generator matrix for the unique self-dual $$[48,24,12]$$[48,24,12] code to see it as a one-generator quasi-cyclic one and obtain six optimal self-orthogonal codes of parameters $$[48 - t, 24 - t, 12]$$[48-t,24-t,12] for $$1 \le t \le 6$$1≤t≤6 with dual distances from 11 to 7 by puncturing the $$[48,24,12]$$[48,24,12] code. Second, a special type of subcode structures for self-orthogonal codes is investigated, and then ten derived dual chains are designed. Third, twelve binary quantum codes are constructed from these derived dual pairs within dual chains using Steane's construction. Ten of them, $$[[42,10,8]], [[44,11,8]], [[45,10,8]], [[45,8,9]], [[46,12,8]]$$[[42,10,8]],[[44,11,8]],[[45,10,8]],[[45,8,9]],[[46,12,8]], $$[[46,9,9]], [[46,5,10]], [[48,13,8]], [[48,9,9]]$$[[46,9,9]],[[46,5,10]],[[48,13,8]],[[48,9,9]], and $$[[48,4,11]]$$[[48,4,11]], achieve as good parameters as the best known ones with comparable lengths and dimensions. Two other codes of parameters $$[[47,4,11]]$$[[47,4,11]] and $$[[48,3,12]]$$[[48,3,12]] are record breaking in the sense that they improve on the best known ones with the same lengths and dimensions in terms of distance.