Prob 4 Consider the systematic 73 RS encoder and decoder The

Prob. 4. Consider the systematic (7,3) R-S encoder and decoder. The received symbols are shown below. R Ca\' a as ao a\' a a (a) Determine the syndrome S. (b) Find the error locator polynomial, o(X) (c) Find the estimated crror polynomial, e(X) (d) Decode the transmitted message.

Solution

Reed-Solomon codes are based on a specialist area of mathematics known as Galois fields or finite fields. A finite field has the property that arithmetic operations (+,-,x,/ etc.) on field elements always have a result in the field. A Reed-Solomon encoder or decoder needs to carry out these arithmetic operations. These operations require special hardware or software functions to implement.

Generator Polynomial

A Reed-Solomon codeword is generated using a special polynomial. All valid codewords are exactly divisible by the generator polynomial. The general form of the generator polynomial is:

and the codeword is constructed using:

c(x) = g(x).i(x)

where g(x) is the generator polynomial, i(x) is the information block, c(x) is a valid codeword and a is referred to as a primitive element of the field.

Generator for RS(255,249)