On the orphans and covering radius of the Reed-Muller codes. (English) Zbl 0767.94008

Langevin, Philippe

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

Keywords:

families of orphans; weight distributions; coset; quadratic form; covering radius