Linear codes of 2-designs associated with subcodes of the ternary generalized Reed–Muller codes

Abstract

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in [C. Ding, C. Li, Infinite families of 2-designs and 3-designs from linear codes, Discrete Mathematics 340(10) (2017) 2415-2431] are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed-Muller codes.

An infinite family of antiprimitive cyclic codes supporting Steiner systems $S(3,8, 7^m+1)$

“…(II) By the conclusions of (I) and Theorem 6, from Theorem 1 we get that both C m and C ⊥ m hold 3-designs. By (12), the number of the codewords with weight q − 7 in C m is A q−7 = 1 336 (q − 1) 2 q(q + 1).…”

“…It is known that linear codes and t-designs are closely related. A t-design can be induced to a linear code (see, for example, [12], [13]). Meanwhile, a linear code C may induce a t-design under certain conditions.…”

The linear codes of t-designs held in the Reed-Muller and Simplex codes

“…Finally, we point out that the idea of using a linear code C 1 supporting a t-design D w (C) to obtain a new linear code C q (D w (C)) may produce a bad or good code. Distance-optimal ternary linear codes were obtained in [11] with this method.…”

Linear Codes Of 2-Designs As Subcodes Of The Extended Generalized Reed-Muller Codes

“…Recently, the 71year-old open problem of the existence of infinity families of linear codes holding 4-designs is settled by Tang and Ding in [12] and it remains open whether there exist infinity families of linear codes holding t-design with t ≥ 5. Ding, Tang and Tonchev [5] studied the linear codes of 2-designs held in a class of affine-invariant ternary codes. The ternary codes used in their paper are defined by quadratic functions.…”