Amir J. Salomon

Augmented product codes and lattices: Reed-Muller codes and Barnes-Wall lattices

This paper concerns the construction of the so-called augmented product codes and augmented product lattices. These are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class, respectively. Certain properties of the augmented product construction are derived, and specific construction examples are given. In particular, it is shown that the Reed-Muller codes, the Golay code, the Barnes-Wall lattices, as well as the Leech lattice all have various augmented product constructions.

Product lattice codes

Lattice-based construction of codes is of theoretical as well as practical importance in particular for communications over bandwidth-limited channels. We investigate the so called product lattice construction. Certain fundamental properties of product lattices are derived, and it is demonstrated that their performance are comparable to those of known good lattices. Practical issues concerning encoding and decoding of product lattices are discussed.

Product Construction of Lattices as Error-Correcting Codes

This letter investigates the so-called product lattice (PL) construction. A PL is obtained from two low-dimensional lattices by means of the Kronecker product. Fundamental properties, mainly those determining the performance of a lattice as an error-correcting scheme over a bandwidth-limited channel, are derived for PLs. Due to their special structure and properties, PLs can provide an attractive family of lattice codes of good performance/complexity tradeoffs

Improved signal superposition coding for cooperative diversity

This paper proposes an improved scheme for cooperative transmit diversity, which applies the Euclidean superposition of modulated signals. Assuming two sources cooperate in transmitting information to a single destination, each source divides its transmit power between locally generated information and relayed information that originated at the other source. This leads to an encoding scheme in which each source transmits the Euclidean superposition of the local information and the relayed information. The new decoding approach at the destination relies on two decoding iterations: First, each transmitted information is evaluated from two consecutive received sequences (once as local information and once as relayed information); second, decoding is performed assuming that the previous and next information sequences were evaluated correctly. Optimization of this approach is shown to be achieved with significantly higher relay power allocation compared to the known approach. It is shown via simulation that this scheme provides significant coding gain compared to the known approach.

Space-Time Block Codes using Diversity Transform

This paper presents a method for constructing space-time block codes for multiple-input multiple-output (MIMO) Rayleigh fading channels. It employs orthogonal designs combined with the so-called diversity transform. Orthogonal designs provide the maximum diversity order for a given number of transmit and receive antennas. The diversity transform relies on unitary transforms that increase the channel alphabet; it does not alter the distance between input sequences, nor the bandwidth or information rate. A scheme of high diversity gain is tailored for any antenna array and signal constellation. Simulation results reveal that this scheme attains higher coding gains than other results known in the art.

Increased diversity space-time coding using the diversity transform

The paper presents a method for constructing space-time block codes for multiple-input multiple-output channels by concatenating orthogonal designs with the so-called diversity transform. Relying on unitary transforms, the diversity transform increases the channel alphabet without sacrificing information rate, bandwidth, or Euclidean distance. The distribution of the resulting channel alphabet is shown to quickly become Gaussian-like. Specific code matrices are constructed and optimized based on the cutoff rate. Both optimum and, reduced-complexity, suboptimum detection algorithms are presented. Simulation results are provided for demonstrating the gains attainable when using the proposed codes.

Space time diagonal codes using lattices

High-rate space-time block coding scheme is constructed for multiple-input multiple-output (MIMO) Rayleigh fading channels. It employs the D-blast transmission scheme combined with lattice constructions tailored for the number of transmit and receive antennas, transmission rate and signal-to-noise ratio. The lattices are constructed with the aid of performance-optimizing criteria, and the obtained coding scheme is rate-preserving, i.e. the bandwidth and information rate of the original D-blast scheme are maintained. The scheme is rich in structure and can therefore provide good tradeoff between coding gain and decoding complexity. Its advantages are more pronounced as the number of antennas increase.

Space-time coding with increased diversity

This paper presents a method for constructing space-time block codes for multiple-input multiple-output (MIMO) Rayleigh fading channels. It employs orthogonal designs combined with the so-called diversity transform. Orthogonal designs provide the maximum diversity order for a given number of transmit and receive antennas. The diversity transform relies on unitary transforms that increase the channel alphabet; it does not alter the distance between input sequences, nor the bandwidth or information rate. A scheme of high diversity gain is tailored for any antenna array and signal constellation. Simulation results reveal that this scheme attains higher coding gains than other results known in the art.

Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2q)

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2q) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2q) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall Λ(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2q) and set partitioning based on partition chains of length-2l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.

Encoding and Decoding Binary Product Lattices

A binary product lattice is generated from two binary component lattices of lower dimensions by employing the Kronecker product. This work focuses on codes carved from binary product lattices. Defined as such, an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points. A novel approach is disclosed yielding an explicit connection between source bits and lattice points. Decoding methods typically used for binary product codes do not apply for product lattices. Several alternative decoding approaches are discussed. In particular, a provably bounded-distance decoder is presented. It relies on the fact that a product lattice code point may be regarded as a two-dimensional array whose rows and columns are points in the component lattices. The obtained results are compared with classical lattices known in the art