Mutually disjoint Steiner systems from BCH codes.

Liu et al. (IEEE Trans Inf Theory 68:3096–3107, 2022) investigated a class of BCH codes C (q , q + 1 , δ , 1) with q = δ m a prime power and proved that the set B δ + 1 of supports of the minimum weight codewords supports a Steiner system S (3 , δ + 1 , q + 1) . In this paper, we give an equivalent formulation of B δ + 1 in terms of elementary symmetric polynomials and then construct a number of mutually disjoint Steiner systems S (3 , δ + 1 , δ m + 1) when m is even and a number of mutually disjoint G-designs G ( δ m + 1 δ + 1 , δ + 1 , δ + 1 , 3) when m is odd. In particular, the existence of three mutually disjoint Steiner systems S (3 , 5 , 4 m + 1) or three mutually disjoint G-designs G ( 4 m + 1 5 , 5 , 5 , 3) is established. [ABSTRACT FROM AUTHOR]