Subquadratic time encodable codes beating the Gilbert-Varshamov bound

We construct explicit algebraic geometry codes built from the Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for alphabet sizes at least 192. Messages are identied with functions in certain Riemann-Roch spaces associated with divisors supported on multiple places. Encoding amounts to evaluating these functions at degree one places. By exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and 1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list) decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent. If \omega = 2, as widely believed, the encoding and decoding runtimes are respectively nearly linear and nearly quadratic. Prior to this work, encoding (resp. decoding) time of code families beating the Gilbert-Varshamov bound were quadratic (resp. cubic) or worse

Efficient Maximum-Likelihood Soft-Decision Decoding of Linear Block Codes Using Algorithm A

In this report we present a novel and efficient maximum-likelihood soft-decision decoding algorithm for linear block codes. The approach used here is to convert the decoding problem into a search problem through a graph which is a trellis for an equivalent code of the transmitted code. Algorithm A*, which uses a priority-first search strategy, is employed to search through this graph. This search is guided by an evaluation function f defined to take advantage of the information provided by the received vector and the inherent properties of the transmitted code. This function f is used to drastically reduce the search space and to make the decoding efforts of this decoding algorithm adaptable to the noise level. Simulation results for the ( 48, 24) and the (72, 36) binary extended quadratic residue codes and the (128, 64) binary extended BCH code are given to substantiate the above claim

A STUDY OF LINEAR ERROR CORRECTING CODES

Since Shannon's ground-breaking work in 1948, there have been two main development streams of channel coding in approaching the limit of communication channels, namely classical coding theory which aims at designing codes with large minimum Hamming distance and probabilistic coding which places the emphasis on low complexity probabilistic decoding using long codes built from simple constituent codes. This work presents some further investigations in these two channel coding development streams. Low-density parity-check (LDPC) codes form a class of capacity-approaching codes with sparse parity-check matrix and low-complexity decoder Two novel methods of constructing algebraic binary LDPC codes are presented. These methods are based on the theory of cyclotomic cosets, idempotents and Mattson-Solomon polynomials, and are complementary to each other. The two methods generate in addition to some new cyclic iteratively decodable codes, the well-known Euclidean and projective geometry codes. Their extension to non binary fields is shown to be straightforward. These algebraic cyclic LDPC codes, for short block lengths, converge considerably well under iterative decoding. It is also shown that for some of these codes, maximum likelihood performance may be achieved by a modified belief propagation decoder which uses a different subset of 7^ codewords of the dual code for each iteration. Following a property of the revolving-door combination generator, multi-threaded minimum Hamming distance computation algorithms are developed. Using these algorithms, the previously unknown, minimum Hamming distance of the quadratic residue code for prime 199 has been evaluated. In addition, the highest minimum Hamming distance attainable by all binary cyclic codes of odd lengths from 129 to 189 has been determined, and as many as 901 new binary linear codes which have higher minimum Hamming distance than the previously considered best known linear code have been found. It is shown that by exploiting the structure of circulant matrices, the number of codewords required, to compute the minimum Hamming distance and the number of codewords of a given Hamming weight of binary double-circulant codes based on primes, may be reduced. A means of independently verifying the exhaustively computed number of codewords of a given Hamming weight of these double-circulant codes is developed and in coiyunction with this, it is proved that some published results are incorrect and the correct weight spectra are presented. Moreover, it is shown that it is possible to estimate the minimum Hamming distance of this family of prime-based double-circulant codes. It is shown that linear codes may be efficiently decoded using the incremental correlation Dorsch algorithm. By extending this algorithm, a list decoder is derived and a novel, CRC-less error detection mechanism that offers much better throughput and performance than the conventional ORG scheme is described. Using the same method it is shown that the performance of conventional CRC scheme may be considerably enhanced. Error detection is an integral part of an incremental redundancy communications system and it is shown that sequences of good error correction codes, suitable for use in incremental redundancy communications systems may be obtained using the Constructions X and XX. Examples are given and their performances presented in comparison to conventional CRC schemes

Design of tch-type sequences for communications

This thesis deals with the design of a class of cyclic codes inspired by TCH codewords. Since TCH codes are linked to finite fields the fundamental concepts and facts about abstract algebra, namely group theory and number theory, constitute the first part of the thesis. By exploring group geometric properties and identifying an equivalence between some operations on codes and the symmetries of the dihedral group we were able to simplify the generation of codewords thus saving on the necessary number of computations. Moreover, we also presented an algebraic method to obtain binary generalized TCH codewords of length N = 2k, k = 1,2, . . . , 16. By exploring Zech logarithm’s properties as well as a group theoretic isomorphism we developed a method that is both faster and less complex than what was proposed before. In addition, it is valid for all relevant cases relating the codeword length N and not only those resulting from N = p

Developing Efficient Algorithms of Decoding the Systematic Quadratic Residue Code with Lookup Tables

The lookup table methods for decoding binary systematic Quadratic Residue (QR) code are presented in this paper. The key ideas behind this decoding technique are based on one to one corresponding mapping between the syndromes and the correctable error patterns. Such algorithms determine the error locations directly by lookup tables without the operations of addition and multiplication over a finite field. Moreover, the methods to dramatically reduce the memory requirement by shift-search decoding are utilized. Two new algorithm have been verified through a software simulation in C language. The new approach is modular, regular and naturally suitable for System on Chip (SOC) software implementation

Fast secure comparison for medium-sized integers and its application in binarized neural networks

In 1994, Feige, Kilian, and Naor proposed a simple protocol for secure 3-way comparison of integers a and b from the range [0, 2]. Their observation is that for p=7, the Legendre symbol (x∣p) coincides with the sign of x for x=a−b∈[−2,2], thus reducing secure comparison to secure evaluation of the Legendre symbol. More recently, in 2011, Yu generalized this idea to handle secure comparisons for integers from substantially larger ranges [0, d], essentially by searching for primes for which the Legendre symbol coincides with the sign function on [−d,d]. In this paper, we present new comparison protocols based on the Legendre symbol that additionally employ some form of error correction. We relax the prime search by requiring that the Legendre symbol encodes the sign function in a noisy fashion only. Practically, we use the majority vote over a window of 2k+1 adjacent Legendre symbols, for small positive integers k. Our technique significantly increases the comparison range: e.g., for a modulus of 60 bits, d increases by a factor of 2.8 (for k=1) and 3.8 (for k=2) respectively. We give a practical method to find primes with suitable noisy encodings.We demonstrate the practical relevance of our comparison protocol by applying it in a secure neural network classifier for the MNIST dataset. Concretely, we discuss a secure multiparty computation based on the binarized multi-layer perceptron of Hubara et al., using our comparison for the second and third layers.</p

An efficient combination between Berlekamp-Massey and Hartmann Rudolph algorithms to decode BCH codes

In digital communication and storage systems, the exchange of data is achieved using a communication channel which is not completely reliable. Therefore, detection and correction of possible errors are required by adding redundant bits to information data. Several algebraic and heuristic decoders were designed to detect and correct errors. The Hartmann Rudolph (HR) algorithm enables to decode a sequence symbol by symbol. The HR algorithm has a high complexity, that's why we suggest using it partially with the algebraic hard decision decoder Berlekamp-Massey (BM). In this work, we propose a concatenation of Partial Hartmann Rudolph (PHR) algorithm and Berlekamp-Massey decoder to decode BCH (Bose-Chaudhuri-Hocquenghem) codes. Very satisfying results are obtained. For example, we have used only 0.54% of the dual space size for the BCH code (63,39,9) while maintaining very good decoding quality. To judge our results, we compare them with other decoders

A low-complexity soft-decision decoding architecture for the binary extended Golay code

International audienceThe (24, 12, 8) extended binary Golay code is a well-known rate-1/2 short block-length linear error-correcting code with remarkable properties. This paper investigates the design of an efficient low-complexity soft-decision decoding architecture for this code. A dedicated algorithm is introduced that takes advantage of the code’s properties to simplify the decoding process. Simulation results show that the proposed algorithm achieves close to maximum-likelihood performance with low computational cost. The decoder architecture is described, and VLSI synthesis results are presented

Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm