n, distance-n/2 code, with 2n codewords, can be formed. Golay code. The Golay code is a (23,12) triple-error-correcting (TEC) code. Constant-ratio codes. Also known as m-out-of-n codes, these have blocklength n, and each codeword has m bits set. These are nonlinear codes of primarily historical interest. Shortened Codes no convenient code exists with exactly the required value of the dataword length, k, a code having a larger value of k, say K', can be shortened. The number of check (parity) bits is unaltered by shortening, but the dataword and codeword lengths are both reduced by the same amount, SO the (n',K') code becomes (n' + k - k',k). An example would be the use of a (15,11) Hamming code, shortened to (12,8), for the byte-by-byte transmission of data. Shortening may or may not increase the minimum distance, and hence the error control ability, of the code. At its simplest, shortening the code consists simply of encoding k' databits, of which k are the required data and k' - k are 0. The main advantage arises when a systematic code is used, because then the first k' - k bits in the codeword can be set to 0 and need not be trans- mitted. In principle, the received codeword can then be lengthened to n' bits again by the reinsertion of k' - k Os prior to decoding with the (n',k') decoder. Chapter 4 Coding In practice, however, the decoder should be modified because errors cannot occur in untransmitted bits. Consider a shortened Hamming code. The Hamming code can normally correct any single error. However, if on receiving a shortened codeword the decoder calcu- lates a syndrome that would normally imply an error in one of the untransmitted bits, that cannot be the true cause. The next most likely cause is a double-error pattern in the trans- mitted bits. If there were only one double-bit error pattern that could produce that syn- drome, the decoder could apply the corresponding double-bit correction. If, however, more than one double-bit error pattern could have produced that syndrome, the decoder cannot guaran