Construction of large cyclic subspace codes with dimension $ k+1 $ via Sidon spaces

Shuhui Yu; Lijun Ji

doi:10.3934/amc.2025057

Article Contents

2026, Volume 22: 48-62. Doi: 10.3934/amc.2025057

This volume Previous Article The construction of two classes of self-orthogonal cyclic codes Next Article Encoding to Legendre curve, Kummer Line and twisted Edwards curve

Construction of large cyclic subspace codes with dimension $ k+1 $ via Sidon spaces

Shuhui Yu^1, , and
Lijun Ji^2,

1.
Department of Mathematics and Physics, Suzhou Vocational University, Suzhou 215104, China
2.
Department of Mathematics, Soochow University, Suzhou 215006, China

^*Corresponding author: Shuhui Yu
^*Corresponding author: Shuhui Yu

Received: June 2025

Revised: October 2025

Early access: November 26, 2025

Published online: November 26, 2025

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Abstract

Subspace codes have attracted much attention in recent decades due to their applications in random network coding. Among these, cyclic subspace codes can be encoded and decoded more efficiently because of their nice algebraic structure, attracting a great deal of research over the past few decades. In this paper, we construct several kinds of $ (k+1) $-dimensional Sidon spaces and present a family of cyclic subspace codes with minimum distance $ 2k $ and size $ q^{k}(2q^{k(r-3)}-1)(q^{n}-1) $, where $ n = (2r+1)k $ and $ r\geq 4 $. These codes are based on our Sidon spaces. Under the same conditions, our codes have larger size than the known ones in the literature.

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Mathematics Subject Classification: Primary: 94B15; Secondary: 11T71.

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Table 1. The lower bounds of cyclic subspace codes in $ \mathcal{G}_{q}(n, k+1) $

$ n $	Sizes	References
$ mk, m\geq 7 $	$ q^{k}\frac{q^{n}-1}{q-1} $	[14, Theorem 3.11]
$ m_{1}=r_{1}k(r_{1}\geq 2) $, $ m_{2}=r_{2}m_{1}(r_{2}>2) $, $ r_{2}\mid r_{1} $	$ (\lceil \frac{m_{1}}{2k} \rceil-1)(\lceil \frac{m_{2}}{2m_{1}} \rceil-1)q^{k}\frac{q^{n}-1}{q-1} $	[16, Theorem 3]
$ \frac{m}{k}\geq 2\ell, \frac{n}{m}\geq 2t $, $ \rho=\lceil\frac{m}{2k} \rceil-1, \theta=\lceil\frac{n}{2m} \rceil-1 $	$ C_{\rho}^{\lfloor \frac{\rho}{2}\rfloor}C_{\theta}^{\lfloor \frac{\theta}{2}\rfloor}\frac{q^{k}(q^{n}-1)}{q-1} $	[18, Theorem 4.1]
$ (2r+1)k, r\geq 4 $	$ 2(r-2)q^{k}(q^{n}-1) $	[28, Theorem 3.7]
$ (2r+1)k, r \geq 4 $	$ q^k(2q^{k(r-3)}-1)(q^{n}-1) $	Theorem 3.3

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