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In this paper we establish the connections between two diierent extensions of Z4-linearity for binary Hamming spaces. We present both notions – propelinearity and G-linearity – in the context of isometries and group actions, taking the viewpoint of geometrically uniform codes extended to discrete spaces. We show a double inclusion relation: binary G-linear… (More)

Let A = (a ij) k×n be a matrix with entries in the Galois field GF (2), and let x = (x 1 , x 2 ,. .. , x n) t 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(H k) = 2 k+1 − 2k − 2, where H k is… (More)

An algebraic decoding algorithm for the expurgated quadratic residue code of length 41 is presented. The algorithm is guaranteed to produce the correct error-location polynomial whenever an error pattern of weight up to four occurs. An error pattern of weight five is not cor-rectable if it is equidistant from the all-zero codeword and a codeword of weight… (More)

In this work we describe a procedure to construct finite signal constellations from lattices associated to rings of algebraic integers and their ideals. The procedure provides a natural way to label the constellation points by elements of a finite field. The labeling is proven to be linear which allows, at the receiver, a fast way to map a constellation… (More)