The theory of error-correcting codes☆

Combinatorial t-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a t-design. Till now only a small amount of work on constructing t-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of 2-designs, and explicitly determine their parameters. The obtained designs cover some earlier 2-designs as special cases. Furthermore, we confirm Conjecture 3 in Ding and Tang (ArXiv:1903.07375, 2019).

Combinatorial t-designs from quadratic functions

Recently, subfield codes of some optimal linear codes have been studied. In this paper, we further investigate a class of subfield codes and generalize the results of the subfield codes of the conic codes in Ding and Wang (Finite Fields Appl. 56, 308–331, 2020). The weight distributions of these subfield codes and the parameters of their duals are determined. Some of the presented codes are optimal or almost optimal according to Grassl (2020) and their duals are distance-optimal with respect to the Sphere Packing bound if p > 3. As a byproduct, we directly obtain the weight distributions of the punctured codes, which is the same with the results presented in Du et al. (2019a, b), and determine the parameters of the duals of the punctured codes. These dual codes are distance-optimal with respect to the Sphere Packing bound with rare exceptions.

A class of subfield codes of linear codes and their duals

/pdf/new-infinite-families-of-near-mds-codes-holding-t-designs-3arr7j11.pdf

New infinite families of near MDS codes holding t-designs

Infinite families of 2-designs derived from affine-invariant codes

Interplay between coding theory and combinatorial t-designs has attracted a lot of attention. It is well known that the supports of all codewords of a fixed Hamming weight in a linear code may hold a t-design. In this paper, we first settle the weight distributions of two classes of linear codes, and then determine the parameters of infinite families of 2-designs held in these codes.

Infinite families of 2-designs from linear codes

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Infinite families of 2-designs from linear codes