The construction of constant dimension subspace codes (CDCs) of larger sizes has gained significant attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximum possible cardinality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A_q(n, d, k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a CDC for given parameters n, d, k and q. This paper introduces a new class of CDCs by improving the inserting construction with the help of the echelon-Ferrers construction for CDCs. Further, we combine this new class of CDCs with those obtained from the parallel linkage construction in order to obtain CDCs of larger size. Our construction method improves some new lower bounds of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_q(n, d, k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, such as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A_q(14, 6, 7), A_q(16, 6, 8), A_q(18, 6, 9)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>14</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>18</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>9</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_q(16, 8, 8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>16</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Improving the inserting construction via echelon-Ferrers construction for constant dimension subspace codes

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发表日期：

2026年2月20日

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Kanchan Singh,
Sheo Kumar Singh

摘要

The construction of constant dimension subspace codes (CDCs) of larger sizes has gained significant attention due to their applications in random network coding. The basic problem of CDCs is to determine the maximum possible cardinality $A_q(n, d, k)$ $A_{q} (n, d, k)$ of a CDC for given parameters n, d, k and q. This paper introduces a new class of CDCs by improving the inserting construction with the help of the echelon-Ferrers construction for CDCs. Further, we combine this new class of CDCs with those obtained from the parallel linkage construction in order to obtain CDCs of larger size. Our construction method improves some new lower bounds of $A_q(n, d, k)$ $A_{q} (n, d, k)$ , such as $A_q(14, 6, 7), A_q(16, 6, 8), A_q(18, 6, 9)$ $A_{q} (14, 6, 7), A_{q} (16, 6, 8), A_{q} (18, 6, 9)$ and $A_q(16, 8, 8)$ $A_{q} (16, 8, 8)$ .