SOME RESULTS IN THE THEORY OF ARITHMETIC CODES,
ILLINOIS UNIV URBANA COORDINATED SCIENCE LAB
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The paper presents a simple number-theoretic investigation of the structure of binary arithmetic AN codes. The range 0, B-1 of represented integers is related to the code length n through 2 to the nth power-1 AB. The analysis is based on the partition of the integers 1 or N or B-1 into orbits, which are analogous to cosets of the multiplicative subgroup of the powers of 2 modulo B. It is shown how the code minimum weight is related to the members of the orbit. The properties of sets of prime powers are used in developing a simple search strategy for codes. An important consequence of the presented analysis is the construction of codes of moderate distance and high rate, thereby filling the spectrum between the two known extremes of the single-error correcting Brown codes and of the maximum-sequence-like codes of Barrows and Mandelbaum. A list of codes of length or 36 is finally presented.