<p>In this article, we construct two classes of mixed-alphabet quasi-cyclic codes over finite rings. The first is a family of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}_2\mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive quasi-cyclic codes of index <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(l_1, l_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and block length <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(l_1m,\, l_2m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mi>m</mi> <mo>,</mo> <mspace width="0.166667em" /> <msub> <mi>l</mi> <mn>2</mn> </msub> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>m</i> is an odd positive integer. The second is a family of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-double quasi-cyclic codes of index <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(l_1, l_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and block length <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(l_1m_1,\, l_2m_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="0.166667em" /> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$m_1, m_2,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and 4 are pairwise coprime positive integers. We study their algebraic structures and determine the generator matrices for codes in these classes. Using a probabilistic approach, we establish their asymptotic properties. For any positive real number <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\delta < \frac{l_1 + l_2}{2(l_1 + 2l_2)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo><</mo> <mfrac> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, and 2-ary entropy <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$h_2\!\left( \frac{(l_1 + 2l_2)\delta }{l_1 + l_2}\right) < \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mn>2</mn> </msub> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mi>δ</mi> </mrow> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </mfrac> </mfenced> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the rate of random <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {Z}_2\mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>-additive quasi-cyclic codes converges to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\frac{1}{l_1 + 2l_2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>l</mi> <mn>2</mn> </msub> </mrow> </mfrac> </math></EquationSource> </InlineEquation>. Similarly, for any positive real number <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\delta < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and 4-ary entropy <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$h_4\!\left( \frac{\delta (l_1m_1 + l_2m_2)}{2(l_1 + l_2)}\right) < \frac{1}{4}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mn>4</mn> </msub> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <mfrac> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>m</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the rate of random <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-double quasi-cyclic codes converges to <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\frac{1}{l_1m_1 + l_2m_2}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </mfrac> </math></EquationSource> </InlineEquation>. In both cases, the relative minimum distance converges to at least <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>. Therefore, both classes of codes are asymptotically good.</p>

Asymptotically good mixed-alphabet quasi-cyclic codes

期刊论文
发表日期：

2026年6月15日

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Asif Khan,
Sajjad Hossain,
Habibul Islam

摘要

In this article, we construct two classes of mixed-alphabet quasi-cyclic codes over finite rings. The first is a family of $\mathbb {Z}_2\mathbb {Z}_4$ $Z_{2} Z_{4}$ -additive quasi-cyclic codes of index $(l_1, l_2)$ $(l_{1}, l_{2})$ and block length $(l_1m,\, l_2m)$ $(l_{1} m, l_{2} m)$ , where m is an odd positive integer. The second is a family of $\mathbb {Z}_4$ $Z_{4}$ -double quasi-cyclic codes of index $(l_1, l_2)$ $(l_{1}, l_{2})$ and block length $(l_1m_1,\, l_2m_2)$ $(l_{1} m_{1}, l_{2} m_{2})$ , where $m_1, m_2,$ $m_{1}, m_{2},$ and 4 are pairwise coprime positive integers. We study their algebraic structures and determine the generator matrices for codes in these classes. Using a probabilistic approach, we establish their asymptotic properties. For any positive real number $\delta < \frac{l_1 + l_2}{2(l_1 + 2l_2)}$ $δ < \frac{l_{1} + l_{2}}{2 (l_{1} + 2 l_{2})}$ , and 2-ary entropy $h_2\!\left( \frac{(l_1 + 2l_2)\delta }{l_1 + l_2}\right) < \frac{1}{2}$ $h_{2} (\frac{(l_{1} + 2 l_{2}) δ}{l_{1} + l_{2}}) < \frac{1}{2}$ , the rate of random $\mathbb {Z}_2\mathbb {Z}_4$ $Z_{2} Z_{4}$ -additive quasi-cyclic codes converges to $\frac{1}{l_1 + 2l_2}$ $\frac{1}{l_{1} + 2 l_{2}}$ . Similarly, for any positive real number $\delta < 1$ $δ < 1$ , and 4-ary entropy $h_4\!\left( \frac{\delta (l_1m_1 + l_2m_2)}{2(l_1 + l_2)}\right) < \frac{1}{4}$ $h_{4} (\frac{δ (l_{1} m_{1} + l_{2} m_{2})}{2 (l_{1} + l_{2})}) < \frac{1}{4}$ , the rate of random $\mathbb {Z}_4$ $Z_{4}$ -double quasi-cyclic codes converges to $\frac{1}{l_1m_1 + l_2m_2}$ $\frac{1}{l_{1} m_{1} + l_{2} m_{2}}$ . In both cases, the relative minimum distance converges to at least $\delta $ $δ$ . Therefore, both classes of codes are asymptotically good.