<p>We introduce homothetic-BCH codes. These are a family of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$q^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-ary classical codes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> of length <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda n_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> are suitable positive integers such that the punctured code <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> in the last <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda n_1 -n_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> coordinates is a narrow-sense BCH code of length <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>n</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. We prove that whenever <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> is Hermitian self-orthogonal, so is <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl <CitationRef CitationID="CR16">2025</CitationRef>). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.</p>

New quantum codes from homothetic-BCH codes

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2025年12月18日

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C. Galindo,
F. Hernando,
H. Martín-Cruz

摘要

We introduce homothetic-BCH codes. These are a family of $q^2$ $q^{2}$ -ary classical codes $\mathcal {C}$ $C$ of length $\lambda n_1$ $λ n_{1}$ , where $\lambda $ $λ$ and $n_1$ $n_{1}$ are suitable positive integers such that the punctured code $\mathcal {B}$ $B$ of $\mathcal {C}$ $C$ in the last $\lambda n_1 -n_1$ $λ n_{1} - n_{1}$ coordinates is a narrow-sense BCH code of length $n_1$ $n_{1}$ . We prove that whenever $\mathcal {B}$ $B$ is Hermitian self-orthogonal, so is $\mathcal {C}$ $C$ . As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths that cannot be reached by BCH codes. With this procedure, we get new quantum codes according to Grassl’s table (Grassl 2025). To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.