Double error Correcting Codes with Improved Code Rates

Designs, Codes and Cryptography, 2012

A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.

IEE Proceedings E Computers and Digital Techniques, 1988

Berlekamp 's key equation needed to decode a Reed-Solomon (RS) code. In this article, a simplified procedure is developed and proved. to correct erasures as well as errors by replacing the initial condition of the Euclidean algorithm by the erasure locator polynomial and the Forney syndrome polynomial. By this means, the errata locator polynomial and the errata evaluator polynomial can be obtained, simultaneously and simply, by the Euclidean algorithm only. With this improved technique the complexity of time-domain RS decoders for correcting both errors and erasures is reduced substantially from previous approaches. As a consequence, decoders for correcting both errors and erasures of RS codes can be made more modular, regular, simple, and naturally suitable for both VLSI and software implementation, A n example illustrating this modified decoding procedure is given for a (I 5, 9) RS code. Recently, Eastman' showed that the errata evaluator polynomial can be computed directly by initializing Berlekamp's

Journal of Complexity, 1997

We present a randomized algorithm which takes as input n distinct points f(x ; y )g from F 2 F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., y = f (x ) for at least t values of i), provided t = ( p nd). The running time is bounded by a polynomial in n. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code.

Encoding and decoding are very important blocks in Communication .There are many techniques to implement Error Detection and Correction Code (EDAC), like Bose Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. These codes can cope with multiple faults, and are based on finite-field arithmetic, also known as Galois Field. BCH codes can correct a given number of bits at any position, whereas RS codes group the bits in blocks to correct them afterwards. RS is a very popular error correcting code, and has been applied to several situations, especially in communication systems. Therefore an efficient error control code is needed to protect the digital data. In high speed communication system Reed-Solomon codes are widely used to provide error protection especially against the burst errors. This is project aims at designing RS encoder and decoder. The RS (n ,k) is chosen as (23,19).This is standard block code which is widely popular in communication .The encoder takes 19 bytes date block and generate 23 byte code block to be transmitted on digital communication channel .This code defines Galois Field GF(2^5) and has the capability of correcting up to 2 short bursts of errors. The encoder and decoder coding is done in VHDL on Xilinx tool .This process is implemented on Xilinx Spartan FPGA.

IEEE Transactions on Information Theory, 1988

In this paper, we first investigate the distance structure of cyclic codes of composite length. A lower bound on the minimum distance for this class of codes is derived. In many cases, the lower bound gives the true minimum distance of a code. Then, we investigate the distance structure of the direct sum of two cyclic codes of composite length. We show that, under certain conditions, the direct-sum code provides two levels of error correcting capability, and hence is a two-level unequal error protection (UEP) code. Finally, a class of two-level UEP cyclic direct-sum codes and a decoding algorithm for a subclass of these codes are presented. ' at most n-1 which implies that a nonzero v(X) has at most n-1 distinct roots. Thus, v(X)=O. This implies that vl(X)=v2(X).

IEEE Transactions on Information Theory, 1992

Error control codes are widely used to increase the reliability of transmission of information over various forms of communications channels. The Hamming weight of a codeword is the number of nonzero entries in the word; the weights of the words in a linear code determine the error-correcting capacity of the code. The r th generalized Hamming weight for a linear code C, denoted by d r (C), is the minimum of the support sizes for r-dimensional subcodes of C. For instance, d 1 (C) equals the traditional minimum Hamming weight of C. In 1991, Feng, Tzeng, and Wei proved that the second generalized Hamming weight d 2 (C) = 8 for all double-error correcting BCH(2 m , 5) codes. We study d 3 (C) and higher Hamming weights for BCH(2 m , 5) codes by a close examination of the words of weight 5. end:

Journal of Electrical Engineering, 2019

Recently a new family of error control codes was proposed which are equivalent to five times extended Reed-Solomon codes. In this paper an erasure decoding algorithm for these codes is proposed.

Information Theory, IEEE Transactions on, 1993

A new approach for encoding any string of information bits into a sequence having bounded running digital sum is presented. The results improve previously known values of the running digital sum for the same rate. Also discussed are ways of incorporating an error-correcting capability into these codes. Some general constructions are given and tables are constructed for specific cases

2008 IEEE International Symposium on Information Theory, 2008

Generalized Concatenated codes are a code construction consisting of a number of outer codes whose code symbols are protected by an inner code. As outer codes, we assume the most frequently used Reed-Solomon codes; as inner code, we assume some linear block code which can be decoded up to half its minimum distance. Decoding up to half the minimum distance of Generalized Concatenated codes is classically achieved by the Blokh-Zyablov-Dumer algorithm, which iteratively decodes by first using the inner decoder to get an estimate of the outer code words and then using an outer error/erasure decoder with a varying number of erasures determined by a set of precalculated thresholds. In this paper, a modified version of the Blokh-Zyablov-Dumer algorithm is proposed, which exploits the fact that a number of outer Reed-Solomon codes with average minimum distanced can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyondd/2. This allows to skip a number of decoding iterations on the one hand and to reduce the complexity of each decoding iteration significantly -while maintaining the decoding performance -on the other.

IEEE Transactions on Information Theory, 2021

Insertion and deletion (insdel for short) errors are synchronization errors in communication systems caused by the loss of positional information in the message. Reed-Solomon codes have gained a lot of interest due to its encoding simplicity, well structuredness and list-decoding capability [7] in the classical setting. This interest also translates to the insdel metric setting, as the Guruswami-Sudan decoding algorithm [7] can be utilized to provide a deletion correcting algorithm in the insdel metric [23]. Nevertheless, there have been few studies on the insdel error-correcting capability of Reed-Solomon codes. Our main contributions in this paper are explicit constructions of two families of 2-dimensional Reed-Solomon codes with insdel error-correcting capabilities asymptotically reaching those provided by the Singleton bound. The first construction gives a family of Reed-Solomon codes with insdel error-correcting capability asymptotic to its length. The second construction provides a family of Reed-Solomon codes with an exact insdel error-correcting capability up to its length. Both our constructions improve the previously known construction of 2-dimensional Reed-Solomon codes [31] whose insdel error-correcting capability is only logarithmic on the code length.

This paper is on the analysis of Reed-Solomon (RS) Codes as an efficient code for error detection and correction. Reed-Solomon codes are examples of error correcting codes in which redundant information is added to data so that it can be recovered reliably despite errors in transmission or storage and retrieval. They describe a systematic way of building codes that can detect and correct multiple errors. It can be used in any communication system where a significant power saving is required and where the operating signal-to-noise ratio is very low. Deep space communications, mobile satellite/cellular communications, microwave links, paging, error correction system used on CD's and DVD's and other communications systems are based on a Reed-Solomon code.

IEEE Transactions on Computers, 1992

In this paper we present the fundamenta. Lheory oft-Error Correcting, k-Error Detecting and d-Unidirectional Error Detecting codes with rl > k > t (t-ECIk-EDId-UED codes). We give a family of methods for designing systematic t-EClk-EDld-UED codes, with d > k > t , and we reveal the methods which give the more efficient, with respect to redundancy, codes for the various values off, k, and d. Also we present the error detection and correction algorithm.

2013

Abstract. In this paper, we consider self-dual codes over the finite ring Zps of integer modulo p s for any prime p and for an integer s ≥ 4. We start with any self-dual code in lower modulo and give an necessary and sufficient condition for the self-duality of induced codes. Then we can give an inductive algorithm for construction of all self-dual codes and the mass formula in case of odd prime p. 1

2012 Information Theory and Applications Workshop, 2012

It is shown that, for all prime powers q and all k ≥ 3, if n ≥ (k − 1)q k − 2 q k −q q−1 , then there exists an [n, k; q] code that is proper for error detection.