The method developed by T. Fujiwara et al. (1985) for efficiently computing the minimum distance of shortened Hamming codes using the weight distribution of their dual codes is extended to treat arbitrary shortened cyclic codes. Using this method implemented on a high-speed special-purpose processor, several classes of cyclic redundancy-check (CRC) codes with 24 and 32 parity bits are investigated. The CRC codes of each class are known to have the same minimum distance d/sub min.L/ in a certain range L of block lengths n, and within each class that CRC code has been determined the minimum distance of which exceeds d/sub min.L/ up to the largest block length. The d/sub min/ profiles of the resulting codes are presented and compared with the d/sub min/ profiles of recent suggestions of P. Merkey and E. C. Posner (1984), as well as with the d/sub min/ profile of the widely used 32 parity-bit standard code recommended in IEEE-802.>

A parity-check matrix for a q-ary repeated-root cyclic code is derived using the Hasse derivative. Then the minimum distance of a q-ary repeated-root cyclic code is expressed in terms of the minimum distance of a certain simple-root cyclic code. With the help of this result, several binary repeated-root cyclic codes of lengths up to n=62 are shown to contain the largest known number of codewords for their given length and minimum distance. The relative minimum distance d/sub min//n of q-ary repeated-root cyclic codes of rate r>or=R is proven to tend to zero as the largest multiplicity of a root of the generator g(x) increases to infinity. It is further shown that repeated-root cycle codes cannot be asymptotically better than simple-root cyclic codes.>

Optimal binary cyclic redundancy-check codes with 16 parity bits (CRC-16 codes) are presented and compared to those in existing standards for minimum-distance, undetected-error probability on binary symmetric channels (BSCs) and properness. The codes in several cases are seen to be superior at block lengths of practical interest when they are used on low-noise BSCs. The optimum minimum distance obtainable by some CRC-16 codes is determined for all block lengths. For several typical low-noise BSCs the minimum undetected error probability achievable with some CRC-16 codes is given for all block lengths.>