Three Combinatorial Perspectives on Minimal Codes

TL;DR

This paper introduces three combinatorial methods to analyze minimal codes and cutting blocking sets, leading to new bounds, constraints, and explicit constructions in finite geometry and coding theory.

Contribution

It presents three novel combinatorial approaches to study minimal codes, resulting in new bounds, constraints, and explicit constructions in finite geometry.

Findings

Derived new bounds and constraints on minimal code parameters.

Constructed small cutting blocking sets in finite projective spaces.

Provided explicit minimal code constructions with short length.

Abstract

We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-F\"uredi Theorem and the Combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. Applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.