The structure of Reed–Muller codes over a nonprime field

I. N. Tumaikin

Lomonosov Moscow State University, Moscow, Russia

Abstract: It is well known that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. The case of a nonprime field is less studied in terms of equalities and inclusions between Reed–Muller codes and radical powers. In this paper, we prove that Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers and provide necessary and sufficient conditions for inclusions between these codes and the powers of the radical.

UDC: 512.715

 English version: Journal of Mathematical Sciences (New York), 2023, 269:3, 422–441