Infinite Families of 3-Designs and 2-Designs From Almost MDS Codes

Abstract

There is a close relationship between linear codes and t-designs. Through their research on a class of narrow-sense BCH codes, Ding and Tang made a breakthrough by presenting the first two infinite families of near MDS codes holding t-designs with t = 2 or 3. In this paper, we present an infinite family of MDS codes over F2s and two infinite families of almost MDS codes over Fps for any prime p, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over F3s , resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost AMDS codes and their dual codes hold infinite families of 3-designs over Fps for any prime p. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.

Low-degree permutation rational functions over finite fields

“…In case q and Q are powers of 2, the roots in F q of polynomials of the form X Q+1 +X+a has attracted much attention. For instance, the number of such roots is studied in [4][5][6]8,10,11,13,15,16,20,23,[25][26][27][28][29][30]34,36,45,48,49,52,56], and this number has been applied to coding theory [7,30,43,55], APN and related functions in cryptography and combinatorics [3,5,8,9,16,33,41,48,51], division rings and combinatorial designs [2,10,19,31,39,40,50,54,55], cross-correlation of m-sequences [11,25,28,…”

On MDS Codes With Galois Hulls of Arbitrary Dimensions

“…These are the first two infinite families of 1-MDS codes found that can support designs. Immediately after that, there were found some infinite families of 1-MDS codes supporting t-designs (see, e.g., [16], [37], [43], [44]). It must be noticed that Heng et al proposed in [16] a conjecture on infinite families of 1-MDS [q − 1, k, q − k − 1] q codes holding 2-designs for each 3 ≤ k ≤ q − 2 (see Conjecture 10 in this paper).…”