Using methods from the theory of algebraic curves over finite fields, together with recent results on the arithmetic of cubic equations over finite fields, we obtain new upper bounds on the second generalized covering radius of binary primitive triple-error-correcting BCH codes. In particular we introduce the notion of weak second generalized covering radius <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R^{(0)}_2(BCH(3,m))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mi>C</mi> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where BCH(3, m) is the binary primitive triple-error-correcting code of length <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$2^m-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mi>m</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. This accounts almost all 2-dimensional <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {F}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-linear subspaces of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {F}^{n-3m}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">F</mi> </mrow> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mn>3</mn> <mi>m</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. Among other results, we show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R^{(0)}_2(BCH(3,m)) \le 9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mi>C</mi> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation> if m is odd and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m \ge 11$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation> (and if m is even and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$m \ge 20$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>20</mn> </mrow> </math></EquationSource> </InlineEquation>).

On the second generalized covering radius for binary primitive triple-error-correcting BCH codes

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发表日期：

2026年7月22日

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Ferruh Özbudak,
İlknur Öztürk

摘要

Using methods from the theory of algebraic curves over finite fields, together with recent results on the arithmetic of cubic equations over finite fields, we obtain new upper bounds on the second generalized covering radius of binary primitive triple-error-correcting BCH codes. In particular we introduce the notion of weak second generalized covering radius $R^{(0)}_2(BCH(3,m))$ $R_{2}^{(0)} (B C H (3, m))$ , where BCH(3, m) is the binary primitive triple-error-correcting code of length $2^m-1$ $2^{m} - 1$ . This accounts almost all 2-dimensional $\mathbb {F}_2$ $F_{2}$ -linear subspaces of $\mathbb {F}^{n-3m}_2$ $F_{2}^{n - 3 m}$ . Among other results, we show that $R^{(0)}_2(BCH(3,m)) \le 9$ $R_{2}^{(0)} (B C H (3, m)) \leq 9$ if m is odd and $m \ge 11$ $m \geq 11$ (and if m is even and $m \ge 20$ $m \geq 20$ ).