Finite Fields

Finite fields are like scale models of the real number system which contain a finite number of "elements" or numbers.  Assume there are n elements in a finite field.

There is one nonbinary finite field associated with each RS or binary BCH code.  For RS codes, that nonbinary field is used to locate or identify each symbol in a codeword and also used as the symbols in codewords.  For binary BCH codes, the nonbinary field is used only to locate or identify the symbols, and the symbols are bits.

Finite fields exist as long as n is a power of a prime.  The theory of algebraic codes only deals with finite fields that have 2m elements so each element is a set of m bits.

All of the operations defined for real numbers are also defined for finite field elements - multiplication, division, addition and subtraction, etc.  Since the field is finite, all of the results of mathematical operations must be in the field.