Divisors on division algebras and error correcting codes

Divisors on Division Algebras and Error Correcting Codes

Recall that a (linear) code is just a k-subspace C of k, where k is some finite field. The elements of C are referred to as codewords, and the dimension of the code, dim(C), is just the dimension of C as a k-space. The length of the code is the dimension of the ambient space k; that is, the length of C is n. The minimum distance, d(a, b) between two codewords a = (a1, . . . , an) and b = (b1, ....

One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...

Synchronizable Error-Correcting Codes

A new technique for correcting synchronization errors in the transmission of discrete-symbol information is developed. The technique can be applied to any t-additive-error-correcting Bose-Chaudhuri-Hocquenghem code, to provide protection against synchronization errors. The synchronization error is corrected at the first complete received word after the word containing the synchronization error,...

Parallel error correcting codes

Error-Correcting Codes

Transmission of information takes place over “noisy” signals. This is the case in satellite transmissions and storing data. For this reason, it is useful to encode information so that errors can be detected and corrected. For our purposes, we will assume all information to be encoded constists of words of fixed length k using symbols from a fixed alphabet, and all encoded messages are divided i...