Arithmetic codes with large distance

Abstract

Arithmetic codes are error-correcting or detecting codes implemented by ordinary arithmetic operations. Arithmetic codes with large distance, and therefore, capable of multierror correction are constructed. These codes are analogous to the finite field codes corresponding to maximal recurring sequences generated by shift registers whose characteristic polynomial is a primitive polynomial. These arithmetic codes are generated by the recurring sequence formed by the inverse of a prime having two as a primitive root. The distance as well as the redundancy increases with the code length. These codes have large redundancy but may be useful in specialized cases. Since the difference between a cyclic shift of a code word and the code word itself is another code word, a two-level function can be formed and the code used as an acquirable code. They can detect error bursts whose length is half the code length. A generalized burst-error correcting code is constructed and it is pointed out that the above large distance codes may be utilized in the construction of this burst-error code.