Extended Hadamard equivalence

Binary sequences with good autocorrelation properties are widely used in cryptography. If the autocorrelation properties are optimum, then the sequences are called perfect. All recently discovered perfect sequences of period n = 2(k) - 1 are Hadamard equivalent, when k is odd. In this paper we generalise this concept to sequences of period n = 4m - 1, where m is not necessarily a power of 2. Using this notion we show, that the Hall and the Legendre sequences are extended Hadamard equivalent.