The Gate Complexity of Syndrome Decoding of Hamming Codes

Let A = (aij)k×n be a matrix with entries in the Galois field GF (2), and let x = (x1, x2, . . . , xn) be a vector of variables assuming values in GF (2). The gate complexity of A, denoted by C(A), is the minimum number of XOR gates necessary to compute the matrix-vector product Ax. In this paper it is shown that C(Hk) = 2k+1 − 2k − 2, where Hk is the parity-check matrix of the [2k − 1, 2k − k − 1] Hamming code. As a consequence, upper bounds on C(A) for any matrix A = (aij)k×n, one for fixed k and another one for fixed n, are derived. As a interesting application of these results, we give a simple proof that the upper bound on the gate complexity of an n× n matrix is 2n2/ log2 n.