Multispreads

TL;DR

This paper characterizes the parameters of multispreads, which are equivalent to additive one-weight codes over finite fields, providing complete results for certain prime power orders and partial results for others.

Contribution

It offers a detailed characterization of multispreads and additive one-weight codes for specific finite field orders, advancing understanding of their structure and parameters.

Findings

Complete characterization for orders 8, 27, and 16.

Partial characterization for prime-square and prime-cube cases.

Establishes equivalence between multispreads and additive one-weight codes.

Abstract

Additive one-weight codes over a finite field of non-prime order are equivalent to special subspace coverings of the points of a projective space, which we call multispreads. The current paper is devoted to the characterization of the parameters of multispreads, which is equivalent to the characterization of the parameters of additive one-weight codes and, via duality, of additive completely regular codes of covering radius 1 (intriguing sets). We characterize these parameters for the case of the prime-square order of the field and make a partial characterization for the prime-cube case and the case of the fourth degree of a prime, including a complete characterization for orders 8, 27, and 16. Keywords: spreads, multispreads, additive codes, one-weight codes, completely regular codes, intriguing sets