Double Error Correcting BCH Code Research Articles

The two covering radius of the two error correcting BCH code

The $m$-covering radii of codes are natural generalizations of the covering radii of codes. In this paper we analyze the 2-covering radii of double error correcting BCH code. In particular, we show that the 2-covering radius of the double error correcting BCH code is $(n+1)/2$ for sufficiently large $n$.

A Memory based Encoder and Decoder for a (15,7)Double Error Correcting BCH Code

A look-up table based encoder and decoder hardware for a (15,7) double error correcting BCH codes is designed and developed. Relevant algorithms for cyclic encoding and parallel decoding are presented. The attraction of this hardware is that it is simple, fast and elegant.

Uniformly Packed Codes and More Distance Regular Graphs from Crooked Functions

Let i>V and i>W be i>n-dimensional vector spaces over GF(2). A function i>Q : i>V → i>W is called i>crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties: We show that crooked functions can be used to construct distance regular graphs with parameters of a Kasami distance regular graph, symmetric 5-class association schemes similar to those recently constructed by de Caen and van Dam from Kasami graphs, and uniformly packed codes with the same parameters as the double error-correcting BCH codes and Preparata codes.

Association Schemes Related to Kasami Codes and KerdockSets

Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).

On a Class of Constant Weight Codes

For any odd prime power $q$ we first construct a certain non-linear binary code $C(q,2)$ having $(q^2-q)/2$ codewords of length $q$ and weight $(q-1)/2$ each, for which the Hamming distance between any two distinct codewords is in the range $[q/2-3\sqrt q/2,\ q/2+3\sqrt q/2]$ that is, 'almost constant'. Moreover, we prove that $C(q,2)$ is distance-invariant. Several variations and improvements on this theme are then pursued. Thus, we produce other classes of binary codes $C(q,n)$, $n\geq 3$, of length $q$ that have 'almost constant' weights and distances, and which, for fixed $n$ and big $q$, have asymptotically $q^n/n$ codewords. Then we prove the possibility of extending our codes by adding the complements of their codewords. Also, by using results on Artin $L-$series, it is shown that the distribution ofthe $0$'s and $1$'s in the codewords we constructed is quasi-random. Our construction uses character sums associated with the quadratic character $\chi$ of $F_{q^n}$ in which the range of summation is $F_q$. Relations with the duals of the double error correcting BCH codes and the duals of the Melas codes are also discussed.

Binary-decision approach to fast Chien search for software decoding of BCH codes

A fast decoding algorithm for binary double error-correcting BCH codes is presented. A binary-decision approach to the Chien search is introduced to reduce the search time by a factor of two, and hence to increase the decoding speed. This algorithm is suitable for microprocessorbased implementation. An illustrative design of a (128, 112, 2) binary BCH decoder using the 16-bit 8086 microprocessor is presented. Complexity, memory space and decoding time are discussed.