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# Calculating the Syndrome of (255,239) Reed-Solomon Code

IP.com Disclosure Number: IPCOM000146577D
Publication Date: 2007-Feb-16
Document File: 2 page(s) / 143K

## Publishing Venue

The IP.com Prior Art Database

## Abstract

Disclosed is a method that decodes Reed-Solomon (RS) codes using a syndrome calculation. Benefits include using less additions and multiplications in the finite fields than traditional schemes.

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# Calculating the Syndrome of (255,239) Reed-Solomon Code

Disclosed is a method that decodes Reed-Solomon (RS) codes using a syndrome calculation. Benefits include using less additions and multiplications in the finite fields than traditional schemes.

## Background

The RS-codes are the class of alternant codes which possess the property of maximum achievable minimal distance  with a given code length  and rate , (i.e. ). These codes are -ary codes with . In practice, the (255,239) RS code is widely used. This code has the minimal distance  and can correct up to  "byte" errors.

## General Description

The disclosed method decodes RS codes using a syndrome calculation. In the disclosed method,  is the RS codeword, and  is the received word, . Vector  describes the errors that occur during transmission, and the decoding task is to find the vector  observing received vector . If  is the parity-check matrix of the RS code, then the syndrome  can be calculated as

.

The syndrome calculation may be described as a multiplication of received words by a parity-check matrix, or as calculating the value of a polynomial associated with the received word in  points of the finite field. It takes  additions and  multiplications to evaluate the polynomial in one point (see Figure 1).

The disclosed method evaluates the syndrome compo...