The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. A linear code C is called <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation>-linear if C has <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-dimensional hull. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$h_\ell $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mi>ℓ</mi> </msub> </math></EquationSource> </InlineEquation>-linear codes with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Therefore, the objective of this paper is to investigate an interesting but non-trivial problem, which is to study some properties of binary <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-linear codes and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. Furthermore, we study the largest minimum distance <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d_{one}(n,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mrow> <mi mathvariant="italic">one</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> among all binary <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$h_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>h</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-linear [n, k] codes. We determine the largest minimum distances <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d_{one}(n,n-k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mrow> <mi mathvariant="italic">one</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ k\le 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$d_{one}(n,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mrow> <mi mathvariant="italic">one</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$k\le 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$14\le n\le 24$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>14</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation>. We partially determine the exact value of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$d_{one}(n,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mrow> <mi mathvariant="italic">one</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$k=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$25\le n\le 30$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>25</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>30</mn> </mrow> </math></EquationSource> </InlineEquation>.

Characterization and construction of $h_1$-optimal binary linear codes related to LCD codes

期刊论文
发表日期：

2026年7月3日

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Shitao Li,
Minjia Shi,
Jon-Lark Kim

摘要

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key to classify finite projective planes. A linear code C is called $h_\ell $ $h_{ℓ}$ -linear if C has $\ell $ $ℓ$ -dimensional hull. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to $h_\ell $ $h_{ℓ}$ -linear codes with $\ell \ge 1$ $ℓ \geq 1$ . Therefore, the objective of this paper is to investigate an interesting but non-trivial problem, which is to study some properties of binary $h_1$ $h_{1}$ -linear codes and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. Furthermore, we study the largest minimum distance $d_{one}(n,k)$ $d_{one} (n, k)$ among all binary $h_1$ $h_{1}$ -linear [n, k] codes. We determine the largest minimum distances $d_{one}(n,n-k)$ $d_{one} (n, n - k)$ for $ k\le 5$ $k \leq 5$ and $d_{one}(n,k)$ $d_{one} (n, k)$ for $k\le 4$ $k \leq 4$ or $14\le n\le 24$ $14 \leq n \leq 24$ . We partially determine the exact value of $d_{one}(n,k)$ $d_{one} (n, k)$ for $k=5$ $k = 5$ or $25\le n\le 30$ $25 \leq n \leq 30$ .

Characterization and construction of \(h_1\)-optimal binary linear codes related to LCD codes

摘要