A note on n-dimensional Hadamard matrices of order 2t and Reed-Muller codes

For the case where H(v,r,n) denotes an n-dimensional Hadamard matrix of order v whose r-dimensional sections are all r-dimensional Hadamard matrices, and where C(v,r,n) denotes the set of H(v,r,n), the author describes the connections between C(2^t,r,n) and Reed-Muller codes. Let R(r,n) denote the rth-order Reed-Muller code of length 2ⁿ. It is shown that C(2,2,n) is a coset of R(1,n) and that C(2,r,n) is the union of cosets of R(1,n). The author shows that when r>or=3, C(2,r,n) is a subcode of R(r-1,n). Moreover, when s,t>or=1 and m>or=r>or=2, C(2^st,r,m) is shown to be a subcode of C(2^s,m(t-1)+r,mt). This allows one to determine the excesses of the proper H(2,2n,n)´s, to prove that the Hadamard transform can be used to repair a corrupted H(2,2,n) in order n2ⁿ steps and that the usual methods for decoding Reed-Muller codes can be used to correct up to n2^n-r-1 errors in any H(2,r,n).