Non-projective two-weight codes

TL;DR

This work extends the theory of two-weight codes beyond the projective setting by examining non-projective codes through the geometry of two-character multisets in $ ext{PG}(k-1,q)$. It develops a characteristic-function framework to represent multisets via hyperplane data, establishes duality between point- and hyperplane-multiplicity descriptions, and proves that, for non-repetitive, non-full-support codes, the non-zero weights must have the form $w_1=up^f$ and $w_2=(u+1)p^f$ with $p$ the field characteristic. The paper also provides structural results that reduce the parameter search to a canonical pair $(s_0,t_0)$ generating the feasible set $ ext{L}(ar{ ext{M}})$ and delivers extensive enumerations for small binary dimensions, including explicit constructions and non-existence results. Overall, the findings offer a geometric and algorithmic path to classify and construct non-projective two-weight codes, with concrete tables of feasible parameters and connections to known two-weight and divisible-code constructions.

Abstract

It has been known since the 1970's that the difference of the non-zero weights of a projective $\mathbb{F}_q$-linear two-weight has to be a power of the characteristic of the underlying field. Here we study non-projective two-weight codes and e.g.\ show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case.

Theorems & Definitions (35)

• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5
• Proposition 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Proposition 4.5