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The authors give a construction of multiblock space--time block codes from cyclic division algebras with a given nth root of unity as the nonnorm element. The construction uses local class field theory. Multiblock codes with M blocks can achieve an M-fold increase in diversity. Index Terms--Division algebra, multiblock, norm, space--time code.

Binary m-sequences are maximal-length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communication. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. This correspondence addresses a conjecture made by Helleseth in 1976, that if m is a power of 2, then there are no pairs of binary m-sequences with a 3-valued crosscorrelation function. This conjecture is proved under the assumption that the three correlation values are symmetric about -1. Index Terms - Binary m-sequences, cyclic codes, crosscorrelation functions.

Binary m-sequences are maximal length sequences generated by shift registers of length m, that are employed in navigation, radar, and spread-spectrum communications systems, because of their crosscorrelation properties. It is well known that given a pair of distinct m-sequences, the crosscorrelation function must take on at least three values. This correspondence considers crosscorrelation functions that take on exactly three values, and where these values are preferred in that they are small. The main result is a proof of a conjecture made by Sarwate and Pursley in 1980, that if m [equivalent to] 0 (mod 4) then there are no preferred pairs of binary m-sequences. The proof makes essential use of a deep theorem of McEliece that restricts the possible weights that can occur in a binary cyclic code. Index Terms - Binary m-sequences, cyclic codes, crosscorrelation functions.

A conjecture of Niho states that under certain assumptions the Fourier transform of the function Tr([x.sup.d]) on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where d = ([2.sup.tk] + 1)/([2.sup.k] + 1), has a spectrum with at most five values. We present a counterexample to this conjecture, and the theory behind finding it. We use the theory of quadratic forms over [F.sub.2]. Index Terms--5-valued, Fourier, quadratic form, spectrum, Walsh.