Full weight spectrum one-orbit cyclic subspace codes

TL;DR

The paper addresses the problem of classifying full weight spectrum (FWS) one-orbit cyclic subspace codes under the subspace metric. It extends the classical notion of maximum weight spectrum to the cyclic subspace-code setting, employing linear-critical-pair analysis and detailed weight-distribution calculations to obtain a complete classification. The main result proves that an FWS one-orbit code must be Orb$(S)$ with $S$ either a polynomial-basis subspace $\langle 1,\lambda,\ldots,\lambda^{k-1}\rangle_{\mathbb{F}_q}$ subject to specific field-extension constraints, or a quadratic-extension form $\langle 1,\lambda,\ldots,\lambda^{l-1}\rangle_{\mathbb{F}_{q^2}}\oplus \lambda^l\mathbb{F}_q$ under even $n$ and $t=[\mathbb{F}_{q^2}(\lambda):\mathbb{F}_{q^2}]\ge 2l+1$. The results unify two constructive families, provide explicit weight distributions, and establish structural invariants under isometries, offering a complete, practically usable classification with implications for network coding and related algebraic coding theory topics.

Abstract

For a linear Hamming metric code of length n over a finite field, the number of distinct weights of its codewords is at most n. The codes achieving the equality in the above bound were called full weight spectrum codes. In this paper we will focus on the analogous class of codes within the framework of cyclic subspace codes. Cyclic subspace codes have garnered significant attention, particularly for their applications in random network coding to correct errors and erasures. We investigate one-orbit cyclic subspace codes that are full weight spectrum in this context. Utilizing number theoretical results and combinatorial arguments, we provide a complete classification of full weight spectrum one-orbit cyclic subspace codes.

Theorems & Definitions (55)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Remark 2.5
• Corollary 2.6
• proof
• Lemma 3.1