Exact Cardinality And Nonredundant Parametrization Of Character-Polynomial Codes

2026-07-13

AI Summary

This work addresses the non-injectivity of character polynomial codes over extension fields, which causes distinct polynomials to generate identical codewords, thereby introducing inaccuracies in code rate and parameter estimation. By leveraging trace mappings and cyclotomic coset analysis, the study provides the first explicit characterization of the redundancy mechanism inherent in these codes. It precisely computes the cardinality of the code and constructs a redundancy-free family of polynomials that uniquely parameterizes each codeword. This contribution rectifies prevailing misconceptions regarding the algebraic structure of character polynomial codes and yields both accurate coding parameters and a more concise representation framework.

Abstract

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.

Problem

• character-polynomial codes
• redundancy
• code cardinality
• non-injectivity
• trace map

Innovation

• character-polynomial codes
• trace map
• cyclotomic cosets
• code cardinality
• nonredundant parametrization