字符多项式码的精确基数与非冗余参数化
Exact Cardinality And Nonredundant Parametrization Of Character-Polynomial Codes
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中文总结 AI 辅助
研究字符多项式码在扩域上存在冗余的问题，通过迹映射和分圆陪集确定非单射性来源、码的精确基数，构造无冗余参数化的精细多项式族，修正了CP码参数并阐明其代数结构。
中文摘要 AI 辅助
字符多项式码是通过对有限域多项式进行求值，并通过加法特征将结果映射到单位复根来构造的。本文表明，在扩域上，原始多项式族可能存在冗余：不同的多项式可以生成相同的码字。我们通过迹映射和分圆陪集确定了这种非单射性的来源，确定了码的精确基数，并构造了一个无冗余参数化码的精细多项式族。这些结果给出了CP码的修正参数并阐明了它们的代数结构。
英文摘要
Character-polynomial codes are constructed by evaluating finite field polynomials and mapping the results to complex roots of unity through additive characters. This paper shows that, over extension fields, the original polynomial family may contain redundancies: distinct polynomials can generate the same codeword. We identify the source of this non-injectivity through the trace map and cyclotomic cosets, determine the exact code cardinality, and construct a refined polynomial family that parametrizes the code without redundancy. These results give corrected parameters for CP codes and clarify their algebraic structure.
