The construction of optimal linear block errorcorrecting codes is not an easy problem, for this, many studies describe methods for generating good. That is, even when taking into account the multitude of errors possible for multilevel quantum systems, topological quantum error-correction codes employing. The point of error correcting codes is to control the error rate. Let's say you have a Sing...

An interesting class of l inear error-correcting codes has been found by Goppa [3], [4]. This paper presents algebraic decoding algorithms for the Goppa codes. These algorithms are only a little more complex than Berlekamp’s well-known algorithm for BCH codes and, in fact, make essential use of his procedure. Hence the cost of decoding a Goppa code is similar to the cost of decoding a BCH code ...

A linear code C is called a group code if C is an ideal in a group algebra K[G] where K is a ring and G is a finite group. Many classical linear error-correcting codes can be realized as ideals of group algebras. Berman [1], in the case of characteristic 2, and Charpin [2], for characteristic p = 2, proved that all generalized Reed–Muller codes coincide with powers of the radical of the group a...

Locally testable codes are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. A linear code C ⊆ F2 is called sparse if dim(C) = O(log(n)). We say that a code C ⊆ F2 is -biased if all nonzero codewords of C have relative weight in the range ( 1 2 − , 1 2 + ), where may be a function of n. Kaufman and Suda...

A decoding rule is presented which minimizes the probability of symbol error over a time-discrete memoryless channel for any linear error-correcting code when the code words are equiprobable. The complexity of this rule varies inversely with code rate, making the technique particularly attractive for high rate codes. Examples are given for both block and convolutional codes.

We consider a concatenated coding scheme using a single inner code, a single outer code, and a fixed single-trial decoding strategy that maximizes the number of errors guaranteed to be corrected in a concatenated codeword. For this scheme, we investigate whether maximizing the guaranteed error correction rate, i.e., the number of correctable errors per transmitted symbol, necessitates pushing t...

This paper gives two lossless speech compression schemes Hamming correction codes compressor (HCCC) and Systematic Linear Block Code Compressor (SLBC) based on error correcting Hamming code and systematic linear block codes respectively. Hamming codes are a family of linear errorcorrecting codes that can detect up to two-bit errors or correct one-bit errors. Systematic linear block code is any ...

In this paper, we first study the error correction and detection capability of codes for a general transmission system inspired by network error correction. For a given weight measure on the error vectors, we define a corresponding minimum weight decoder. Then we obtain a complete characterization of the capability of a code for 1) error correction; 2) error detection; and 3) joint error correc...

In classical algebraic coding theory, the minimum distance of block code completely determines the ability of the code in terms of error correction/detection and erasure correction. We have obtained generalizations of these results for network codes.