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Nonvanishing determinants have emerged as an attractive criterion enabling a space--time code achieve the optimal diversity-multiplexing gains tradeoff. It seems that cyclic division algebras play the most crucial role in designing a space--time code with nonvanishing determinants. In this paper, we explicitly construct space--time codes for arbitrary numbers of transmit antennas that achieve nonvanishing determinants and the optimal diversity-multiplexing gains tradeoff over Z [i]. Unlike previous methods usually arising a field compositum for two or more fields, our scheme, which only requires one simple extension, constitutes a much more efficient and feasible advancement whether in theory or practice. Index Terms--Field extensions, Galois theory, optimal diversity-multiplexing gains tradeoff, space-time codes.

This correspondence presents new rate-1 space--time block codes (STBCs) attending full diversity over every quadrature amplitude modulation (QAM) constellation when the number of Tx antennas is a power of two. From the simulation results, our design performs very closely to the quasi-orthogonal code with constellation rotation over 4-QAM and 16-QAM in the case of four Tx antennas over quasi-static Rayleigh fading channels. Moreover, the proposed codes would not cause any constellation expansion over QAM symbols in contrast with the quasi-orthogonal codes with constellation rotations. Index Terms--Algebraic codes, generalized quaternion groups, representations of finite groups, space--time block codes (STBCs), transmit diversity.

The design potential of using quaternionic numbers to identify a 4 x 4 real orthogonal space-time block code has been exploited in various communication articles. Although it has been shown that orthogonal codes in full-rate exist only for 2 Tx-antennas in complex constellations, a series of complex quasi-orthogonal codes for 4 Tx-antennas is still proposed to have good performance recently. This quasi-orthogonal scheme enables the codes to reach the optimal nonorthogonality, which can be measured by taking the expectation over all transmit signals of the ratios between the powers of the off-diagonal and diagonal components. This correspondence extends the quaternionic identification to the above encoding methods. Based upon tensor product for giving the quaternionic space a linear extension, a complete necessary and sufficient condition for identifying any given complex quasi-orthogonal code with the extended space is generalized by considering every possible two-dimensional R-algebra. Index Terms--Algebraic codes, division algebras, quasi-orthogonal space--time block code (STBC), quaternions, R-algebras, representations of finite groups, tensor product.

Abstract: Lubin conjectures that for an invertible series to commute with a noninvertible series with only simple roots of iterates, two such commuting power series must be endomorphisms of a single formal group. In this paper, we show that if the reduction of these two commuting power series are endomorphisms of a formal group, then themselves are endomorphisms of a formal group. [Copyright &y& Elsevier]