On the orphans and covering radius of the reed-muller codes

Abstract

In [4] is given an inductive proof of the existence of families of orphans of RM(1, m) whose weight distributions are {2^m−1 − ∈2^(m+k−2)/2 | ∈ = −1, 0, 1}, where k satisfies 0≤k<m and k ≡ m (mod 2). We show that any coset of RM(1, m) having this kind of distribution is an orphan. In particular, the coset of a not completely degenerate quadratic form is always an orphan. Working about the conjecture which says that the covering radius of RM(1, m) is even, we prove that an orphan of odd weight of RM(1, m) cannot be 0-covered. Finally, we simplify our proof, given in [5], that the distance from any cubic of RM(3, 9) to RM(1, 9) is at most 240.