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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=2k–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.