差错控制编码解决加性噪声 第15页

差错控制编码解决加性噪声 第15页

and so coresponds to a codeword of .
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Remark:There will be a unique polynomial such that . When thinking of the codespace as a product of polynomials with instead of the row space of a binary matrix, this polynomial will take the place of the check matrix in the decoding process. Let be a codeword then

Then if degthe coeffiecients of the highest powers match the coeffiecients of the lowest powers of x in the polynomial .

Further, if the generating polynomial has degree , then the codewords form a basis for the codespae with a generating matrix

and we can realize a code word as an information sequence times the matrix

mod

As stated before, we can treat the polynomial , having the condition that mod , as the check polynomial of the code . Knowing that we want for , it follows that the check matrix for the code is

Lets now consider an example using a matrix of 0's and 1's.

Because each of the three rows are linearly independent we know from linear algebra that the row space of the matrix is a 3-dimensional subspace of where is the field . In fact, since the algorithm is based on left multiplication by a codevector, the row space is the range of the matrix. In essence, because each codevector corresponds to a polynomial, the codespace corresponds to cyclic shifts of the coefficients of polynomials, which is just all polynomial products where is the generating polynomial for the code .

It turns out that in this case all the cyclic shifts of the first row vector account for every vector in the row space. Thus we have a cyclic code and the generating polynomial is

which corresponds to the first row of the generating matrix.

Also, and so the check polynomial is

and we get the following check matrix

Now taking the codeword corresponding to we get

mod

In fact, for every codeword in the space , the product will be the zero vector. This will be an extremely important detail when creating an algorithm for a single-error correcting BCH code.

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