linear code, and in the classical encoding procedure, its generator matrix is . Systematic encoding procedure: The message as an initial sequence of values There is an alternative encoding procedure that also produces the Reed–Solomon code, but that does so in a systematic way. Here, the mapping from the message to the polynomial works differently: the polynomial is now defined as the unique polynomial of degree less than such that To compute this polynomial from , one can use Lagrange interpolation. Once it has been found, it is evaluated at the other points of the field. The alternative encoding function for the Reed–Solomon code is then again just the sequence of values: Since the first entries of each codeword coincide with , this encoding procedure is indeed systematic. Since Lagrange interpolation is a linear transformation, is a linear mapping. In fact, we have , where Discrete Fourier transform and its inverse A discrete Fourier transform is essentially the same as the encoding procedure; it uses the generator polynomial p(x) to map a set of evaluation points into the message values as shown above: The inverse Fourier transform could be used to convert an error free set of n < q message values back into the encoding polynomial of k coefficients, with the constraint that in order for this to work, the set of evaluation points used to encode the message must be a set of increasing powers of α: However, Lagrange interpolation performs the same conversion without the constraint on the set of evaluation points or the requirement of an error free set of message values and is used for systematic encoding, and in one of the steps of the Gao decoder. The BCH view: The codeword as a sequence of coefficients In this view, the message is interpreted as the coefficients of a polynomial . The sender computes a related polynomial of degree where and sends the polynomial . The polynomial is constructed by multiplying the message polynomial , which has degree , with a generator polynomial of degree that is known to both the sender and the receiver. The generator polynomial is defined as the polynomial whose roots are sequential powers of the Galois field primitive For a "narrow sense code", . Systematic encoding procedure The encoding procedure for the BCH view of Reed–Solomon codes can be modified to yield a systematic encoding procedure, in which each codeword