Orthogonal Designs and a Cubic Binary Function

Orthogonal designs are fundamental mathematical notions used in the construction of space time block codes for wireless transmissions. Designs have two important parameters, the rate and the decoding delay; the main problem of the theory is to construct designs maximizing the rate and minimizing the decoding delay. All known constructions of CODs are inductive or algorithmic. In this paper, we present an explicit construction of optimal CODs. We do not apply recurrent procedures and do calculate the matrix elements directly. Our formula is based on a cubic function in two binary n-vectors. In our previous work (Comm. Math. Phys., 2010, and J. Pure and Appl. Algebra, 2011), we used this function to define a series of non-associative algebras generalizing the classical algebra of octonions and to obtain sum of squares identities of Hurwitz-Radon type

Four-Group Decodable Space-Time Block Codes

Two new rate-one full-diversity space-time block codes (STBC) are proposed. They are characterized by the \emph{lowest decoding complexity} among the known rate-one STBC, arising due to the complete separability of the transmitted symbols into four groups for maximum likelihood detection. The first and the second codes are delay-optimal if the number of transmit antennas is a power of 2 and even, respectively. The exact pair-wise error probability is derived to allow for the performance optimization of the two codes. Compared with existing low-decoding complexity STBC, the two new codes offer several advantages such as higher code rate, lower encoding/decoding delay and complexity, lower peak-to-average power ratio, and better performance.Comment: 1 figure. Accepted for publication in IEEE Trans. on Signal Processin

Space Frequency Codes from Spherical Codes

A new design method for high rate, fully diverse ('spherical') space frequency codes for MIMO-OFDM systems is proposed, which works for arbitrary numbers of antennas and subcarriers. The construction exploits a differential geometric connection between spherical codes and space time codes. The former are well studied e.g. in the context of optimal sequence design in CDMA systems, while the latter serve as basic building blocks for space frequency codes. In addition a decoding algorithm with moderate complexity is presented. This is achieved by a lattice based construction of spherical codes, which permits lattice decoding algorithms and thus offers a substantial reduction of complexity.Comment: 5 pages. Final version for the 2005 IEEE International Symposium on Information Theor

Structure theorem of square complex orthogonal design

Square COD (complex orthogonal design) with size $[n, n, k]$ is an $n \times n$ matrix $\mathcal{O}_z$, where each entry is a complex linear combination of $z_i$ and their conjugations $z_i^*$, $i=1,\ldots, k$, such that $\mathcal{O}_z^H \mathcal{O}_z = (|z_1|^2 + \ldots + |z_k|^2)I_n$. Closely following the work of Hottinen and Tirkkonen, which proved an upper bound of $k/n$ by making a crucial observation between square COD and group representation, we prove the structure theorem of square COD

Low-delay, High-rate Non-square Complex Orthogonal Designs

The maximal rate of a non-square complex orthogonal design for $n$ transmit antennas is $1/2+\frac{1}{n}$ if $n$ is even and $1/2+\frac{1}{n+1}$ if $n$ is odd and the codes have been constructed for all $n$ by Liang (IEEE Trans. Inform. Theory, 2003) and Lu et al. (IEEE Trans. Inform. Theory, 2005) to achieve this rate. A lower bound on the decoding delay of maximal-rate complex orthogonal designs has been obtained by Adams et al. (IEEE Trans. Inform. Theory, 2007) and it is observed that Liang's construction achieves the bound on delay for $n$ equal to 1 and 3 modulo 4 while Lu et al.'s construction achieves the bound for $n=0,1,3$ mod 4. For $n=2$ mod 4, Adams et al. (IEEE Trans. Inform. Theory, 2010) have shown that the minimal decoding delay is twice the lower bound, in which case, both Liang's and Lu at al.'s construction achieve the minimum decoding delay. % when $n=2$ mod 4. For large value of $n$, it is observed that the rate is close to half and the decoding delay is very large. A class of rate-1/2 codes with low decoding delay for all $n$ has been constructed by Tarokh et al. (IEEE Trans. Inform. Theory, 1999). % have constructed a class of rate-1/2 codes with low decoding delay for all $n$. In this paper, another class of rate-1/2 codes is constructed for all $n$ in which case the decoding delay is half the decoding delay of the rate-1/2 codes given by Tarokh et al. This is achieved by giving first a general construction of square real orthogonal designs which includes as special cases the well-known constructions of Adams, Lax and Phillips and the construction of Geramita and Pullman, and then making use of it to obtain the desired rate-1/2 codes. For the case of 9 transmit antennas, the proposed rate-1/2 code is shown to be of minimal-delay.Comment: To appear in IEEE Transactions on Information Theor