<p>Projective Reed–Muller (PRM) codes, defined via evaluation of homogeneous polynomials over projective space, constitute an important class of error-correcting codes. This paper investigates the hulls of PRM codes. Firstly, we extend results of Kaplan and Kim by computing the hull dimension of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$PRM(q, m, v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi>R</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in four new parameter ranges, including resolution of their open problem for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$q - 1< v < \frac{3(q - 1)}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo><</mo> <mi>v</mi> <mo><</mo> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, and provide explicit formulas. Secondly, we determine the minimum distance of the hull for all nontrivial parameters. Thirdly, we analyze the hull of a class of PRM codes with nonstandard dual properties, showing that for parameters in the range <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{m(q-1)}{2}< v < m(q-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mrow> <mi>m</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> <mo><</mo> <mi>v</mi> <mo><</mo> <mi>m</mi> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$v \equiv 0 \pmod {q-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the hull is precisely another Projective Reed–Muller code.</p>

Hull parameters of Projective Reed–Muller codes

期刊论文
发表日期：

2026年2月20日

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Yufeng Song,
Jinquan Luo

摘要

Projective Reed–Muller (PRM) codes, defined via evaluation of homogeneous polynomials over projective space, constitute an important class of error-correcting codes. This paper investigates the hulls of PRM codes. Firstly, we extend results of Kaplan and Kim by computing the hull dimension of $PRM(q, m, v)$ $P R M (q, m, v)$ in four new parameter ranges, including resolution of their open problem for $q - 1< v < \frac{3(q - 1)}{2}$ $q - 1 < v < \frac{3 (q - 1)}{2}$ , and provide explicit formulas. Secondly, we determine the minimum distance of the hull for all nontrivial parameters. Thirdly, we analyze the hull of a class of PRM codes with nonstandard dual properties, showing that for parameters in the range $\frac{m(q-1)}{2}< v < m(q-1)$ $\frac{m (q - 1)}{2} < v < m (q - 1)$ with $v \equiv 0 \pmod {q-1}$ $v \equiv 0 (mod q - 1)$ , the hull is precisely another Projective Reed–Muller code.