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<p>In this paper, we consider a special class of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((\bar{\lambda },\theta ,\ell )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-monomial codes over finite fields, where we describe the Galois duals of one-generator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((\bar{\lambda },\theta ,\ell )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-monomial codes generated by a generator of the form <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((g(x), g(x)f_1(x), \ldots , g(x)f_{\ell -1}(x))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>f</mi> <mrow> <mi>ℓ</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We give necessary and sufficient conditions to obtain the Galois self-orthogonality and Galois LCD properties. Furthermore, we construct certain maximum-distance-separable quantum error-correcting codes (MDS QECCs) using the CSS construction from Euclidean and Hermitian self-orthogonal codes. Similarly, we utilize LCD codes to construct certain maximum-distance-separable entanglement-assisted quantum error-correcting codes (MDS EAQECCs).</p>

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Galois self-orthogonal and Galois LCD one-generator \((\bar{\lambda },\theta ,\ell )\)-monomial codes

  • 期刊论文

  • 发表日期：

    2025年11月3日

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  • Oussama Kabbouch,
  • Lhousain Mouatadid,
  • Mustapha Najmeddine,
  • Nuh Aydin

摘要

In this paper, we consider a special class of \((\bar{\lambda },\theta ,\ell )\) ( λ ¯ , θ , ℓ ) -monomial codes over finite fields, where we describe the Galois duals of one-generator \((\bar{\lambda },\theta ,\ell )\) ( λ ¯ , θ , ℓ ) -monomial codes generated by a generator of the form \((g(x), g(x)f_1(x), \ldots , g(x)f_{\ell -1}(x))\) ( g ( x ) , g ( x ) f 1 ( x ) , … , g ( x ) f ℓ - 1 ( x ) ) . We give necessary and sufficient conditions to obtain the Galois self-orthogonality and Galois LCD properties. Furthermore, we construct certain maximum-distance-separable quantum error-correcting codes (MDS QECCs) using the CSS construction from Euclidean and Hermitian self-orthogonal codes. Similarly, we utilize LCD codes to construct certain maximum-distance-separable entanglement-assisted quantum error-correcting codes (MDS EAQECCs).