f codeword may be successfully recovered if the decoder a Galois field must be a power of a prime. Conversely, for decodes the received word to the closest codeword and the any prime power 9, one can construct a Galois field of amount of noise is less than dmin/2. Note that if the block size q. Let GF(q) be a finite field of size q and let n be code is linear, then the minimum distance dmin is simply a positive integer. Then it is easy to check that [GF(q)]" the smallest weight of a nonzero codeword. Here the is a linear space over GF(q). An (n, k) linear block code weight of a codeword is the number of its nonzero symbols. C over GF(q) is a k-dimensional linear subspace of the For the Hamming distance, successful decoding linear space [GF(q)]". In particular, for any two code- translates into there being less than dmin/2 changes to the words C1, C2 E C, the sum of the codewords is also symbols of the original codeword, where dmin represents a codeword, C1 C. Since C is a k-dimensional linear the minimum Hamming distance between any pair of space, we can find a set of k basis vectors SO that every codewords. For the Euclidean distance, successful codeword can be expressed as a linear combination of the decoding translates into the magnitude of the noise being basis vectors. In vector-matrix notation, we can express less than dmin/2. For the aforementioned, we may every codeword C of C as visualize the correct decoding region of each codeword to be a sphere with radius dmin/2 (as in Figure 3.4-2), and c=uG, (3.4.2) thus if a codeword is corrupted by noise, as long as the noise does not perturb the codeword to be outside of its where u is a 1 X k vector of field elements and Gisakxn correct decoding region, then successful decoding will be matrix whose k rows are k basis vectors. The matrix G is guaranteed. It is apparent that for a given n and k, it is known as a generator matrix and elementary row desirable to find a code that maximizes dmin. operations can be performed on G to form another matrix Once a code wit