Abstract: The generating polynomials of higher degree residue codes over finite fields are factors of the polynomial[xn-1]. Generally speaking, it is difficult to factor the polynomial[xn-1] over finite fields. This paper gives generating idempotents of cubic and quartic residue codes over the field [F2]. As a result, the generating polynomials of cubic and quartic residue codes over the field [F2] can be obtained by computing the greatest common divisors of these generating idempotents and the polynomial[xn-1] with computer software such as Matlab and Maple。

Key words: generating idempotent, residue code, cyclic code

摘要： 有限域上高次剩余码的生成多项式都是多项式[xn-1]的因式。针对多项式[xn-1]在有限域上分解的困难性，给出了二元域[F2]上三次和四次剩余码的幂等生成元表达式。利用计算机软件求解该幂等生成元与[xn-1]最大公因式就可得到三次和四次剩余码生成多项式而不用分解[xn-1]。

关键词: 幂等生成元, 剩余码, 循环码