🤖 AI Summary
This paper addresses the parameter characterization of additive one-weight codes over finite fields of non-prime order, which is equivalent to classifying multispreads in projective spaces and additive completely regular codes of covering radius one—also known as “interesting sets.” Employing an integrated approach combining combinatorial design theory, finite geometry, and coding theory, the authors systematically analyze subspace structures over finite fields $mathbb{F}_q$ of prime power order. They fully determine all feasible parameters for $q = p^2$, and provide optimal or complete parameter classifications for $q = p^3$ and $p^4$, including the cases $q = 8, 16, 27$. The work establishes a rigorous one-to-one correspondence between multispreads and additive one-weight codes, thereby advancing the structural theory of additive completely regular codes with covering radius one.
📝 Abstract
Additive one-weight codes over a finite field of non-prime order are equivalent to special subspace coverings of the points of 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. 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).