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

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## 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.

## Keywords

Quadratic Form Boolean Function Binary Code Covering Radius Inductive Proof
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## 10 Bibliography

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1991