Short minimal codes and covering codes via strong blocking sets in projective spaces

Abstract

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension k over the q-element Galois field which is linear in both q and k, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.

Higgledy-piggledy sets in projective spaces of small dimension

“…Proof. The lower bound on m(5, q) is proven in [2, Theorem 2.14] for general k, and reproven geometrically in [20,Theorem 3.9]. This lower bound can in fact be improved by 1 if q 9 ([2, Corollary 2.19]).…”

“…As one generally wishes to construct higgledy-piggledy sets of small size, lower bounds on the size of such sets were determined to reveal which sizes would (theoretically) be optimal. A lower bound on higgledy-piggledy line sets was determined in [15] for q large enough, and very recently strengthened in [20] to all values of q.…”

Small Strong Blocking Sets by Concatenation

“…or some shorter ones constructed in [2], whose length is quadratic in the dimension. Even if we cannot get strong blocking sets smaller than those in [21], this construction has the great advantage of being explicit.…”

“…It is worth mentioning that minimal codes form a class of asymptotically good codes [1,14]. Families of short minimal codes (via their equivalence with strong blocking sets) were provided in [5,21] also in connection with line spreads or sets of lines in higgledy-piggledy arrangement [19]. Apart from a few small dimensional cases, none of these constructions were effective.…”