tee to correct the error. In decoders for shortened cyclic codes, further modifications are required (see Shorten- ing Cyclic Codes section). Extended Codes An odd-distance code can 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 distance is 2t + 1, the code is t-error correcting; the extended code can correct up to t errors and detect the case in which there are t + 1 errors, which is often useful. An example of an extended code is the (24,12) extension of the Golay (23,12) code, which has a convenient rate, k/n, of exactly 0.5. Interleaved and Concatenated Codes Interleaving (also known as interlacing) is one technique used to combat burst errors, such as those that occur in fading radio channels. As an illustration, consider arranging 110 databits as ten 11-bit rows and encoding each row using a Hamming (15,11) coder. Assume that the resulting codewords are stored as the ten rows of a 10 X 15 bit array, but that the array is then transmitted column by column. This means that the first bit of each codeword will be transmitted, then the second of each, and SO on. The receiver regroups the received bits into the original pattern of ten 15-bit codewords before decoding. In this example, if there is a burst error of up to ten bits duration, no more than one bit in each codeword will be affected. Because the code is single-error correcting, the receiver can correct these errors, SO the use of 10-way interleaving allows correction of 10-bit burst errors. The extension of this technique to more powerful codes is straightforward and permits correction of combinations of random and burst errors. Concatenated coding means the application of one error-correcting code to the input data, followed by the application of another code to codewords output by the first coder, and SO on. Concatenated coding is valuable when the different codes can combat different types of error (e.g., burst versus random). It