The problem to construct optimal conflict-avoiding codes of even lengths and the Hamming weight 3 is completely settled. On the contrary, it is still open for odd lengths. It turns out that the prime lengths are the fundamental cases needed to be constructed. In the article, we study conflict-avoiding codes of prime lengths and give a connection with the so-called cyclotomic numbers. By having some nonzero cyclotomic numbers, a well-known algorithm for constructing optimal conflict-avoiding codes will work for certain prime lengths. As a consequence, we are able to answer the size of optimal conflict-avoiding code for a new class of prime lengths.

When an approximately universal code is designed with cyclic division algebras, the normalized diversity product can be intrinsically increased by a small integer non-norm element. While 2 + i has been the smallest integer non-norm element in all known 8 × 8 designs over QAM, this paper presents another new 8 × 8 code with the smaller non-norm element 1 + i over QAM.

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.

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

It has been shown that a complex orthogonal design that provides full diversity and full transmission rate is not possible for more than two transmit antennas. The paper presents a new class of quasi-orthogonal space-time block codes using a group-theoretic method. The new designs can achieve full diversity for quadrature phase-shift-keying modulation, like the system using rotated constellations. Based upon the analysis of the new codes, a relaxed designing viewpoint for full diversity is proposed for arbitrary signal constellations.

The design potential of using quaternionic numbers to identify a 4/spl times/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.