 to be in the databit in that row and S =ch (14.7) column, which is therefore corrected. If 2 or more row sums, or 2 or 14/6 Coding more column sums, are 1, a larger number of errors has occurred, 14.3.10 Extended codes which cannot be corrected. An odd-distance code may be extended by adding an overall (even) parity bit; for binary codes this is just the modulo-2 sum of the codebits. This increases the code distance by 1. If the original 14.3.8 Other codes distance is 2t + 1, the code is t-error correcting; the extended code can correct up to t errors and detect the case when there are t+1 The most important linear block codes are the cyclic codes, which errors, which is often useful. An example of an extended code is the are considered in Section 14.4. In addition, many other linear and (24,12) extension of the Golay (23,12) code, which has a convenient non-linear block codes have been developed. These include: rate k/n of exactly 0.5. 1. Hadamard codes. The code words are the rows of a Hadamard matrix, which is a binary n X n matrix (n even), in which each 14.4 Interleaved and concatenated codes row differs from any other row in exactly n/2 positions; if the matrix elements are denoted +1 and -1, the rows are also Interleaving (also known as interlacing) is a technique used to orthogonal. Using this matrix and its complement (i.e. the combat burst errors, such as those which occur in fading radio matrix formed by exchanging the +1 and -1 elements), a channels. As an illustration, consider ten 15-bit codewords pro- blocklength n, distance n/2 code, with 2n code words can be duced by a Hamming (15,11) coder. Assume that these are stored as formed. the 10 rows of a 10 X 15 bit array, and that the array is then 2. Golay code. The Golay code is a (23,12) triple error correcting transmitted column by column. This means that the 10 first bits of (TEC) code. the 15 codewords will be transmitted, then the 10 second bits, and Constant ratio codes. Also known as m-out-of-n codes, have so on. The receiver re-groups the re