Reed-Solomon Codes: Irving Reed And Gus Solomon

2.2.1 Reed-Solomon Codes
Irving Reed and Gus Solomon [37] on January 21, 1959, submitted a paper which was published in June 1960 in the Journal of the society for Industrial and Applied mathematics with the title “Polynomial codes over certain finite fields”. This paper introduced a new class of error correcting codes that are now called Reed-Solomon codes.
Reed-Solomon codes[38][39] are constructed and decoded by using finite field arithmetic. Finite fields were the discovery of French mathematician Evariste Galois thus they are also referred to as Galois fields. The finite field has the property that arithmetic operations (+, -, x, / etc.) on field elements always have a result inside the field. They are block-based error correcting codes
Designers are not required to use the natural sizes of Reed-Solomon code blocks. A technique known as "shortening" produces a smaller code of any desired size from a larger code. For example, the widely used (255,251) code can be converted to a (160,128). A Reed–Solomon code operating on 8-bits symbols has n = 28 – 1 = 255 symbols per block because the number of symbol in the encoded block is n = 2m – 1. For the designer, its capability to correct both burst errors makes it the best choice to use as the encoding and decoding
The general idea is the construction of a polynomial, the coefficient produced will be symbols such that the generator polynomial will exactly divide the data/parity polynomial. The basic principle of encoding is to find the remainder for the message divided by a generator polynomial G(x).The Reed-Solomon codeword is generated using a special polynomial known as the Generator polynomial. All valid code words are exactly divisible by the generator polynomial. The generator polynomial of the RS encoder is represented