Since polynomial codes are linear codes, the minimum Hamming distance is equal to the minimum weight of any non-zero codeword.
2.
The algebraic nature of polynomial codes, with cleverly chosen generator polynomials, can also often be exploited to find efficient error correction algorithms.
3.
As for all digital codes, the error detection and correction abilities of polynomial codes are determined by the minimum Hamming distance of the code.
4.
With this definition of the codewords, it can be immediately seen that a Reed Solomon code is a polynomial code, and in particular a BCH code.
5.
For the purposes of constructing polynomial codes, we identify a string of n symbols a _ { n-1 } \ ldots a _ 0 with the polynomial
6.
Note that this, as every polynomial code, is indeed a linear code, i . e ., linear combinations of code words are again code words.
7.
Since the polynomial code is defined over the Binary Galois Field GF ( 2 ) = \ { 0, 1 \ }, polynomial elements are represented as a modulo-q sum and the final polynomials are:
8.
The " polynomial code generated by g ( x ) " is the code whose code words are precisely the polynomials of degree less than n that are divisible ( without remainder ) by g ( x ).
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