Majority-Logic Decoding of Generalized Reed–Muller Codes

We consider the possibility of simple (i.e., when the number of checks increases as $n\log_2 n$ with code length $n$) majority-logic decoding of generalized Reed–Muller codes (GRM codes), defined as various powers of the radical of the group algebra of the group of type $(p,\dots,p)$ over a field of characteristic $p$. A simple majority-logic decoding algorithm realizing the code distance is constructed for first-order $p$-ary GRM codes and for ternary GRM codes of any order.