. A further technique for decoding RS codes is called transform decoding, which uses a finite field analogue of the Fourier transform. A particularly important ability of RS codes (and nonbinary BCH codes in general) is their ability to perform error and erasure decoding. This is of value if the receiver can under some circumstances signal the loss of a received digit. For example, in a CD player the demodulator can often detect invalid (corrupt) digits. If such a code has distance d, it can correct combinations of 1 errors and S erasures provided that 21 + S < d. While RS codes are naturally suited to nonbinary digits, they can sometimes be used very effectively for binary channels by treating groups of m bits as 2"-ary digits. It has been shown that such codes outperform binary codes with the same rate and block length at low output error rates. Also, when used in this way, RS codes have a natural burst error- correcting ability because, for example, a burst of up to m bit errors will affect at most 2 "digits." Convolutional Codes Majority Logic Decodable Codes These cyclic codes are slightly inferior to BCH codes in terms of error correction, but they have simple decoder implementations in which the error pattern detector circuit is a com- bination of XOR and majority logic gates. Burst Error-Correcting Codes Fire codes are a widely used class of algebraically constructed burst error-correcting cyclic codes, which require a minimum of 3b - 1 parity bits to correct bursts of up to b bits in length. Other, more efficient burst error-correcting cyclic codes, found by computer search, are also available. Shortening Cyclic Codes Cyclic codes can be shortened; the design of a systematic cyclic coder is unaffected by this, except that only k bits are clocked in rather than k' before the control line is switched to output the parity check bits. In the decoder, the same approach of simply reducing the number of clocks can be used for syndrome polynomial calculation; however, if error cor- rection is to be carried out, 