Polynomial code

In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial). In coding theory, a polynomial code is a type of linear code whose set of valid code words consists of those polynomials (usually of some fixed length) that are divisible by a given fixed polynomial (of shorter length, called the generator polynomial). Fix a finite field G F ( q ) {displaystyle GF(q)} , whose elements we call symbols. For the purposes of constructing polynomial codes, we identify a string of n {displaystyle n} symbols a n − 1 … a 0 {displaystyle a_{n-1}ldots a_{0}} with the polynomial Fix integers m ≤ n {displaystyle mleq n} and let g ( x ) {displaystyle g(x)} be some fixed polynomial of degree m {displaystyle m} , called the generator polynomial. The polynomial code generated by g ( x ) {displaystyle g(x)} is the code whose code words are precisely the polynomials of degree less than n {displaystyle n} that are divisible (without remainder) by g ( x ) {displaystyle g(x)} . Consider the polynomial code over G F ( 2 ) = { 0 , 1 } {displaystyle GF(2)={0,1}} with n = 5 {displaystyle n=5} , m = 2 {displaystyle m=2} , and generator polynomial g ( x ) = x 2 + x + 1 {displaystyle g(x)=x^{2}+x+1} . This code consists of the following code words: