<p>The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to check the permutation equivalence of two linear codes and compute a linear code’s automorphism group. Research has shown that these algorithms are very effective when the hull size is small. Linear complementary dual (LCD) codes have the smallest hulls, while codes with a one-dimensional hull have the second smallest. A recent notable paper that directs our investigation is authored by H. Chen, titled “On the Hull-Variation Problem of Equivalent Linear Codes", published in IEEE Transactions on Information Theory, volume 69, issue 5, in 2023. In this paper, we first explore the one-dimensional hull of a linear code over finite fields. Additionally, we demonstrate that any LCD code over an extended binary field <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mathrm{I\!F}_q $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="normal">I</mi> <mspace width="-0.166667em" /> <mi mathvariant="normal">F</mi> </mrow> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> (where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ q > 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>) with a minimum distance of at least 2 is equivalent to the one-dimensional hull of a linear code under a specific weak condition. Furthermore, we provide a construction for creating hulls with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \ell + 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>-dimensionality from an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-dimensional hull of a linear code, again under a weak condition. This corresponds to a particularly challenging direction, as creating <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-dimensional hulls from <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ \ell + 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>-dimensional hulls. Finally, we derive several constructions for the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-dimensional hulls of linear codes as a consequence of our results.</p>

On construction of linear (Euclidean) hull codes over finite extensions binary fields

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2025年12月20日

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Sanjit Bhowmick,
Deepak Kumar Dalai,
Sihem Mesnager

摘要

The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to check the permutation equivalence of two linear codes and compute a linear code’s automorphism group. Research has shown that these algorithms are very effective when the hull size is small. Linear complementary dual (LCD) codes have the smallest hulls, while codes with a one-dimensional hull have the second smallest. A recent notable paper that directs our investigation is authored by H. Chen, titled “On the Hull-Variation Problem of Equivalent Linear Codes", published in IEEE Transactions on Information Theory, volume 69, issue 5, in 2023. In this paper, we first explore the one-dimensional hull of a linear code over finite fields. Additionally, we demonstrate that any LCD code over an extended binary field $ \mathrm{I\!F}_q $ ${I F}_{q}$ (where $ q > 3 $ $q > 3$ ) with a minimum distance of at least 2 is equivalent to the one-dimensional hull of a linear code under a specific weak condition. Furthermore, we provide a construction for creating hulls with $ \ell + 1 $ $ℓ + 1$ -dimensionality from an $ \ell $ $ℓ$ -dimensional hull of a linear code, again under a weak condition. This corresponds to a particularly challenging direction, as creating $ \ell $ $ℓ$ -dimensional hulls from $ \ell + 1 $ $ℓ + 1$ -dimensional hulls. Finally, we derive several constructions for the $ \ell $ $ℓ$ -dimensional hulls of linear codes as a consequence of our results.