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<p>Partitions of the binary linear Hamming code into Preparata-like codes are known to induce line-parallelisms of PG(<i>n</i>,&#xa0;2). In this paper, we show that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> is any Preparata-like code contained in the binary linear Hamming code <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of the same length, then <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> can be partitioned into additive translates of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(P_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. This generalizes a result of Baker, van Lint, and Wilson who prove this fact for the class of generalized Preparata codes. We give an explicit description for line-parallelisms obtained from such a partition via crooked Preparata-like codes and establish an equivalence criterion for such line-parallelisms.</p>

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Line-parallelisms of PG(n, 2) from Preparata-like codes

  • 期刊论文

  • 发表日期：

    2026年7月15日

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  • Philipp Heering,
  • Vladislav Taranchuk

摘要

Partitions of the binary linear Hamming code into Preparata-like codes are known to induce line-parallelisms of PG(n, 2). In this paper, we show that if \(P_n\) P n is any Preparata-like code contained in the binary linear Hamming code \(H_n\) H n of the same length, then \(H_n\) H n can be partitioned into additive translates of \(P_n\) P n . This generalizes a result of Baker, van Lint, and Wilson who prove this fact for the class of generalized Preparata codes. We give an explicit description for line-parallelisms obtained from such a partition via crooked Preparata-like codes and establish an equivalence criterion for such line-parallelisms.