The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this paper, we suggest the other type of Hamiltonian formalism for quantum error correction code witho...

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.

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...

Use of an error correction code in a given transmission channel can be regarded as the statistical experiment. Therefore, powerful results from the theory of comparison of experiments can be applied to compare the performances of different error correction codes. We present results on the comparison of block error correction codes using the representation of error correction code as a linear ex...

Lomont, Chris C.. Ph.D., Purdue University, May, 2003. Error Correcting Codes on Algebraic Surfaces. Major Professor: Tzuong-Tsieng Moh. Error correcting codes are defined and important parameters for a code are explained. Parameters of new codes constructed on algebraic surfaces are studied. In particular, codes resulting from blowing up points in P are briefly studied, then codes resulting fr...

It is shown that the quantum error-correction can be acheived by the using of classical binary codes or additive codes over F4 (see [1],[2],[3]). In this paper with the help of some algebraic techniques the theory of algebraic-geometric codes is used to construct asymptotically good family of quantum error-correcting codes and other classes of good quantum error-correcting codes. Our results ar...

Error correction codes are used for long years to protect memories from the soft errors. For a single bit error correction, the SEC single bit error correction code that correct one bit error per word are used. Double bit error detection code are used to detect the double bit errors. In the increasing of the technology the single bit error correction codes are used in the various places such as...