# zbMATH — the first resource for mathematics

On the orphans and covering radius of the Reed-Muller codes. (English) Zbl 0767.94008
Applied algebra, algebraic algorithms and error-correcting codes, Proc. 9th Int. Symp., AAECC-9, New Orleans/LA (USA) 1991, Lect. Notes Comput. Sci. 539, 234-240 (1991).
Summary: [For the entire collection see Zbl 0758.00016.]
R. A. Brualdi, N. Cai and V. S. Pless have given an inductive proof of the existence of families of orphans of $$RM(1,m)$$ whose weight distributions are $$\{2^{m-1}-\varepsilon 2^{(m+k- 2)/2}\mid\varepsilon=-1,0,1\}$$, where $$k$$ satisfies $$0\leq k<m$$ and $$k\equiv m\pmod 2$$. 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 show that the distance from any cubic of $$RM(3,9)$$ to $$RM(1,9)$$ is at most 240.

##### MSC:
 94B05 Linear codes, general