Phase detection based on a single parity check code for low SNR applications

The improvement of carrier phase synchronization through the use of channel decoding information has been recently widely studied. However, an initial synchronization step is necessary prior to the first decoding iteration. When the phase to estimate is not constant (due to frequency error or phase noise), this first step is problematic on relatively long coded words. In this paper we propose to add a short single parity-check code to the transmission scheme for the synchronization purpose. The proposed phase detector is based on the soft symbol decoding of the additional code. Our simulation results show that the proposed synchronization scheme outperforms the classical non data-aided and decision directed algorithms for continuous transmissions with 8PSK modulation, in presence of phase noise. It allows indeed to reduce the phase jitter in considerable proportions. Requiring few additional complexity, it seems well adapted to satellite continuous transmissions.De nombreuses études ont récemment montré l'avantage que l'on peut tirer du décodage canal pour l'amélioration de la synchronisation de phase porteuse à faible rapport signal à bruit. Lorsque la phase à estimer varie au cours du temps (erreur de fréquence, bruit de phase), l'estimation de la phase avant la première étape de décodage pose toutefois problème lorsque les mots de code sont longs. Nous proposons ici un algorithme de synchronisation fondé sur l'introduction d'un code de parité simple et court. Nous montrons que pour une transmission continue et une modulation MDP8, le détecteur proposé permet de réduire sensiblement la gigue de phase à très faible point de fonctionnement et en présence de bruit de phase, par rapport aux algorithmes classiques. De faible complexité, il apparait notamment bien adapté au contexte des télécommunications par satellite

Optimal Linear and Cyclic Locally Repairable Codes over Small Fields

We consider locally repairable codes over small fields and propose constructions of optimal cyclic and linear codes in terms of the dimension for a given distance and length. Four new constructions of optimal linear codes over small fields with locality properties are developed. The first two approaches give binary cyclic codes with locality two. While the first construction has availability one, the second binary code is characterized by multiple available repair sets based on a binary Simplex code. The third approach extends the first one to q-ary cyclic codes including (binary) extension fields, where the locality property is determined by the properties of a shortened first-order Reed-Muller code. Non-cyclic optimal binary linear codes with locality greater than two are obtained by the fourth construction.Comment: IEEE Information Theory Workshop (ITW) 2015, Apr 2015, Jerusalem, Israe

Decomposition Methods for Large Scale LP Decoding

When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper we draw on decomposition methods from optimization theory, specifically the Alternating Directions Method of Multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a "two-slice" characterization of the geometry of the parity polytope, which is the convex hull of all codewords of a single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by ADMM and allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher signal-to-noise ratio than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the 49th Annual Allerton Conference, September 2011. This version to appear in IEEE Transactions on Information Theor

High-Throughput Random Access via Codes on Graphs

Recently, contention resolution diversity slotted ALOHA (CRDSA) has been introduced as a simple but effective improvement to slotted ALOHA. It relies on MAC burst repetitions and on interference cancellation to increase the normalized throughput of a classic slotted ALOHA access scheme. CRDSA allows achieving a larger throughput than slotted ALOHA, at the price of an increased average transmitted power. A way to trade-off the increment of the average transmitted power and the improvement of the throughput is presented in this paper. Specifically, it is proposed to divide each MAC burst in k sub-bursts, and to encode them via a (n,k) erasure correcting code. The n encoded sub-bursts are transmitted over the MAC channel, according to specific time/frequency-hopping patterns. Whenever n-e>=k sub-bursts (of the same burst) are received without collisions, erasure decoding allows recovering the remaining e sub-bursts (which were lost due to collisions). An interference cancellation process can then take place, removing in e slots the interference caused by the e recovered sub-bursts, possibly allowing the correct decoding of sub-bursts related to other bursts. The process is thus iterated as for the CRDSA case.Comment: Presented at the Future Network and MobileSummit 2010 Conference, Florence (Italy), June 201

On some new approaches to practical Slepian-Wolf compression inspired by channel coding

This paper considers the problem, first introduced by Ahlswede and Körner in 1975, of lossless source coding with coded side information. Specifically, let X and Y be two random variables such that X is desired losslessly at the decoder while Y serves as side information. The random variables are encoded independently, and both descriptions are used by the decoder to reconstruct X. Ahlswede and Körner describe the achievable rate region in terms of an auxiliary random variable. This paper gives a partial solution for the optimal auxiliary random variable, thereby describing part of the rate region explicitly in terms of the distribution of X and Y

XJ-BP: Express Journey Belief Propagation Decoding for Polar Codes

This paper presents a novel propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational complexity of MS BP decoding by 90.4%; this enables an energy-efficient hardware implementation of BP decoding in practice.Comment: submitted to GLOBECOMM 201

Describing A Cyclic Code by Another Cyclic Code

A new approach to bound the minimum distance of $q$-ary cyclic codes is presented. The connection to the BCH and the Hartmann--Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum distance in terms of parameters of the associated code