Most versions of the Peterson-Gorenstein-Zierler (PGZ) decoding algorithm are not true bounded distance decoding algorithms in the sense that when a received vector is not in the decoding sphere of any codeword, the algorithm does not always declare a decoding failure. For a t-error-correcting BCH code, if the received vector is at distance i, i t, from a codeword in a supercode with BCH distan...

It is shown that a classical error correcting code C = [n, k, d] which contains its dual, C⊥ ⊆ C, and which can be enlarged to C ′ = [n, k′ > k + 1, d′], can be converted into a quantum code of parameters [[n, k + k′ − n,min(d, d3d′/2e)]]. This is a generalisation of a previous construction, it enables many new codes of good efficiency to be discovered. Examples based on classical Bose Chaudhur...

Recently a framework for assisted quantum error correction was proposed in which a specific type of error is allowed to occur on auxiliary qubits, which is in contrast to standard entanglement assistance that requires noiseless auxiliary qubits. However, while the framework maintains the ability to import any binary or quaternary linear code without sacrificing active error correction power, it...

Absiruct-A linear code C over GF( q) is good for t-error-correction and error detection if P(C, t; c) I P(C, t; (q 1)/q) for all z, 0 s c I (q 1)/q, where P( C, t; c) is the probability of an undetected error after a codeword in C is transmitted over a q-ary symmetric channel with error probability c and correction is performed for ail error patterns with t or fewer errors. A sufficient conditi...

The goal of this article is to provide a gentle introduction to the basic definitions, goals and constructions in coding theory. In particular we focus on the algorithmic tasks tackled by the theory. We describe some of the classical algebraic constructions of error-correcting codes including the Hamming code, the Hadamard code and the Reed Solomon code. We describe simple proofs of their error...

A new technique, based on the pseudo-random properties of certain graphs, known as expanders, is used to obtain new simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling and then regrouping the code coordinates. For any fixed (small) rate, and for sufficiently large alphabet, the codes ...

Low-Density Parity-Check (LDPC) codes have recently drawn much attention due to their nearcapacity error correction performance, and are currently in the focus of many standardization activities, e.g., IEEE 802.11n, IEEE 802.16e, and ETSI DVB-S. In this contribution, we discuss several aspects related to the practical application of such codes to wireless communications systems. We consider fle...

Reliability and security are two major criteria for survivable communications in error-prone wireless environments. To ensure reliable communications, Forward Error Correcting (FEC) codes such as Reed-Solomon (RS) codes are employed for error detection and correction by adding redundancies into the original data to form code words. Secure data communications based on FEC are achieved in many tr...