Achievable Rates for K-user Gaussian Interference Channels

The aim of this paper is to study the achievable rates for a $K$ user Gaussian interference channels for any SNR using a combination of lattice and algebraic codes. Lattice codes are first used to transform the Gaussian interference channel (G-IFC) into a discrete input-output noiseless channel, and subsequently algebraic codes are developed to achieve good rates over this new alphabet. In this context, a quantity called efficiency is introduced which reflects the effectiveness of the algebraic coding strategy. The paper first addresses the problem of finding high efficiency algebraic codes. A combination of these codes with Construction-A lattices is then used to achieve non trivial rates for the original Gaussian interference channel.Comment: IEEE Transactions on Information Theory, 201

A simple algorithm for decoding Reed-Solomon codes and its relation to the Welch-Berlekamp algorithm

A simple and natural Gao algorithm for decoding algebraic codes is described. Its relation to the Welch-Berlekamp and Euclidean algorithms is given.Comment: 7 pages. Submitted to IEEE Transactions on Information Theor

On the Optimality of the Golden Code

In this note, we prove the optimality of the Golden Code inside the class of cyclic algebras based codes. In doing so, we get better insight on these algebraic codes, not only in dimension 2, but more generally for higher dimension, and summarizing the different approaches tried so far to optimize them, we derive design strategies that we believe are the key to either show the optimality of existing codes or give a way to improve them

Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation

For many algebraic codes the main part of decoding can be reduced to a shift register synthesis problem. In this paper we present an approach for solving generalised shift register problems over skew polynomial rings which occur in error and erasure decoding of $\ell$-Interleaved Gabidulin codes. The algorithm is based on module minimisation and has time complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem.Comment: 10 pages, submitted to WCC 201

Stall Pattern Avoidance in Polynomial Product Codes

Product codes are a concatenated error-correction scheme that has been often considered for applications requiring very low bit-error rates, which demand that the error floor be decreased as much as possible. In this work, we consider product codes constructed from polynomial algebraic codes, and propose a novel low-complexity post-processing technique that is able to improve the error-correction performance by orders of magnitude. We provide lower bounds for the error rate achievable under post processing, and present simulation results indicating that these bounds are tight.Comment: 4 pages, 2 figures, GlobalSiP 201

Algebraic Codes For Error Correction In Digital Communication Systems

Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible error-free in the presence of noise. Subsequently the notion of using error correcting codes to mitigate the effects of noise in digital transmission was introduced by R. Hamming. Algebraic codes, codes described using powerful tools from algebra took to the fore early on in the search for good error correcting codes. Many classes of algebraic codes now exist and are known to have the best properties of any known classes of codes. An error correcting code can be described by three of its most important properties length, dimension and minimum distance. Given codes with the same length and dimension, one with the largest minimum distance will provide better error correction. As a result the research focuses on finding improved codes with better minimum distances than any known codes. Algebraic geometry codes are obtained from curves. They are a culmination of years of research into algebraic codes and generalise most known algebraic codes. Additionally they have exceptional distance properties as their lengths become arbitrarily large. Algebraic geometry codes are studied in great detail with special attention given to their construction and decoding. The practical performance of these codes is evaluated and compared with previously known codes in different communication channels. Furthermore many new codes that have better minimum distance to the best known codes with the same length and dimension are presented from a generalised construction of algebraic geometry codes. Goppa codes are also an important class of algebraic codes. A construction of binary extended Goppa codes is generalised to codes with nonbinary alphabets and as a result many new codes are found. This construction is shown as an efficient way to extend another well known class of algebraic codes, BCH codes. A generic method of shortening codes whilst increasing the minimum distance is generalised. An analysis of this method reveals a close relationship with methods of extending codes. Some new codes from Goppa codes are found by exploiting this relationship. Finally an extension method for BCH codes is presented and this method is shown be as good as a well known method of extension in certain cases

Statistical Mechanics of Broadcast Channels Using Low Density Parity Check Codes

We investigate the use of Gallager's low-density parity-check (LDPC) codes in a broadcast channel, one of the fundamental models in network information theory. Combining linear codes is a standard technique in practical network communication schemes and is known to provide better performance than simple timesharing methods when algebraic codes are used. The statistical physics based analysis shows that the practical performance of the suggested method, achieved by employing the belief propagation algorithm, is superior to that of LDPC based timesharing codes while the best performance, when received transmissions are optimally decoded, is bounded by the timesharing limit.Comment: 14 pages, 4 figure