Restrictions on Weight Distribution of Reed-Muller Codes

I t is shown t h a t in the r t h order binary Reed-Muller code of length N = 2" and m i n i m u m distance d = 2~-L the only code-words having weight between d and 2d are those with weights of the form 2d-2 t for some i. The same result also holds for certain super-codes of the R M codes. DEFI~ITm~<. T h e weight of an integer k, W(/~), is the n u m b e r of l's in the binary expansion of ]~. T h e binary R e e d-M u l l e r code is conveniently defined as an extension of a cyclic code. Let a be a primitive element of GF(2"~). DEFINITION. For 1 _-< r =< m-2, the rth order binary Punctured R e e d-M u l l e r code of length 2 m-1 is a cyclic code whose generator polynomial has as roots those a k such t h a t i _-< W(k) _-< m-r-i DEFINITION. For 1 _-< r = m-2, codewords of the rth order binary Reed-5,iuller code of length N = 2 m are formed by adding an overall parity cheek c~ to the codewords [co, cl,. .-, cN_~] of the rth order P u n c t u r e d R e e d-M u l l e r code.