Let N, k be positive integers and let q be a prime power. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {G}_q(N,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">G</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the set of all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {F}}_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation>-subspaces of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathbb {F}}_{q^N}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>q</mi> <mi>N</mi> </msup> </msub> </math></EquationSource> </InlineEquation> with dimension k. In this paper, We propose two classes of optimal multi-orbit cyclic subspace codes using new combinatorial approaches. The first construction enlarges the size of optimal cyclic subspace codes by improving a class of the Sidon spaces presented by Zhang and Ge (J Algebr Comb 55:781–794, 2022). For any positive integers N, k and prime power q such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N\ge 4k-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>4</mn> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$N|(q-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">|</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we establish new multi-orbit cyclic subspace codes in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {G}_q(N,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">G</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with optimal minimum distance <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2k-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The second construction makes use of variants of the Sidon spaces introduced by Roth et al. (IEEE Trans Inf Theory 64(6):4412–4422, 2018). We give a new lower bound for cyclic subspace codes in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {G}_q(N,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">G</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where k is a factor of N and q is a prime power.

Two kinds of optimal multi-orbit cyclic subspace codes via Sidon spaces

期刊论文
发表日期：

2026年6月14日

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He Zhang,
Chunming Tang,
Xiwang Cao,
Guangkui Xu

摘要

Let N, k be positive integers and let q be a prime power. Let $\mathcal {G}_q(N,k)$ $G_{q} (N, k)$ denote the set of all ${\mathbb {F}}_q$ $F_{q}$ -subspaces of ${\mathbb {F}}_{q^N}$ $F_{q^{N}}$ with dimension k. In this paper, We propose two classes of optimal multi-orbit cyclic subspace codes using new combinatorial approaches. The first construction enlarges the size of optimal cyclic subspace codes by improving a class of the Sidon spaces presented by Zhang and Ge (J Algebr Comb 55:781–794, 2022). For any positive integers N, k and prime power q such that $N\ge 4k-1$ $N \geq 4 k - 1$ and $N|(q-1)$ $N | (q - 1)$ , we establish new multi-orbit cyclic subspace codes in $\mathcal {G}_q(N,k)$ $G_{q} (N, k)$ with optimal minimum distance $2k-2$ $2 k - 2$ . The second construction makes use of variants of the Sidon spaces introduced by Roth et al. (IEEE Trans Inf Theory 64(6):4412–4422, 2018). We give a new lower bound for cyclic subspace codes in $\mathcal {G}_q(N,k)$ $G_{q} (N, k)$ , where k is a factor of N and q is a prime power.