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<p>For a code <i>C</i> in a space with maximal distance <i>n</i>, we say that <i>C</i> has symmetric distances if its distance set <i>S</i>(<i>C</i>) is symmetric with respect to <i>n</i>/2. In this paper, we prove that if <i>C</i> is a binary code with length 2<i>n</i>, constant weight <i>n</i> and symmetric distances, then <Equation ID="Equ25"> <EquationSource Format="TEX">\( |C| \le \left( {\begin{array}{c}2 n - 1\\ |S(C)|\end{array}}\right) . \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mo stretchy="false">|</mo> <mi>C</mi> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>2</mn> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </Equation>This result can be interpreted using the language of Johnson association schemes. More generally, we give a framework to study codes with symmetric distances in Q-bipartite Q-polynomial association schemes, and provide upper bounds for such codes. Moreover, we use number theoretic techniques to determine when the equality holds.</p>

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Codes with symmetric distances

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Gábor Hegedüs,
  • Sho Suda,
  • Ziqing Xiang

摘要

For a code C in a space with maximal distance n, we say that C has symmetric distances if its distance set S(C) is symmetric with respect to n/2. In this paper, we prove that if C is a binary code with length 2n, constant weight n and symmetric distances, then \( |C| \le \left( {\begin{array}{c}2 n - 1\\ |S(C)|\end{array}}\right) . \) | C | ≤ 2 n - 1 | S ( C ) | . This result can be interpreted using the language of Johnson association schemes. More generally, we give a framework to study codes with symmetric distances in Q-bipartite Q-polynomial association schemes, and provide upper bounds for such codes. Moreover, we use number theoretic techniques to determine when the equality holds.