In this paper, we first generalize the LCP of codes over finite fields to the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive complementary pair (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-ACP) of codes over the mixed alphabet <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. Then we provide three judging criteria for an <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive code pair (C, D) to be an <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-ACP of codes. Meanwhile, we also give a sufficient condition for an <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive code pair (C, D) to be an <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-ACP of codes. In addition, we exhibit properties and characterizations for an <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive code pair (C, D) to be an <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-additive complementary pair (<InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-ACP) of codes. Finally, we provide an interesting application of an <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathbb {F}_2\mathbb {F}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">F</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-ACP of codes in coding for the two-user binary adder channel.

$\mathbb {F}_2\mathbb {F}_4$-ACP of codes

期刊论文
发表日期：

2026年7月15日

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Xiusheng Liu,
Jie Liu

摘要

In this paper, we first generalize the LCP of codes over finite fields to the $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -additive complementary pair ( $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -ACP) of codes over the mixed alphabet $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ . Then we provide three judging criteria for an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -additive code pair (C, D) to be an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -ACP of codes. Meanwhile, we also give a sufficient condition for an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -additive code pair (C, D) to be an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -ACP of codes. In addition, we exhibit properties and characterizations for an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -additive code pair (C, D) to be an $\mathbb {F}_4$ $F_{4}$ -additive complementary pair ( $\mathbb {F}_4$ $F_{4}$ -ACP) of codes. Finally, we provide an interesting application of an $\mathbb {F}_2\mathbb {F}_4$ $F_{2} F_{4}$ -ACP of codes in coding for the two-user binary adder channel.

\(\mathbb {F}_2\mathbb {F}_4\)-ACP of codes

摘要