This paper presents a new class of AN codes of high efficiency capable of correcting single errors in radix-r arithmetic with the binary coded digits—BCr—system. The errors corrected within a single radix-r digit are single errors of the binary digits with weight ±wi≤ r–1,i=1,2,···, m, which are used to encode the BCr digits. The corresponding arithmetic unit is made out of slices, one for each BCr digit. The AN codes considered have a generator A of the form A = τ·p where τ, p odd and rp=n-1 ≤ τk± 1. The criteria are applied to the BCD system, and we determine all τ < 100 and p < 200 for the most important BCD codes with weights ± wi < r –1, i = 1, 2, 3, 4. For each BCD code, the paper gives the numbers τ < 100 for which AN codes exist, and the maximum code efficiency E = 2·m·k/(A-1) attained at some p < 200.

This note is concerned with the design of fast totally self-checking checkers for m-out-of-(2m ± 1) codes. The new method uses only three levels of gates, and is based on the partitioning of the input lines into m blocks of two lines per block except for the last block. A first level of two-input AND and OR gates realizes the majority functions T(ki ≥ 1) and T(ki ≥ 2). These are combined through AND gates of a second level into the so-called product functions Pj1j2...jm, one for each class of input codewords that have jl, j2,···, jm 1's in the corresponding blocks of the input lines. Finally, two OR gates (third level) partition the product functions into two. The property of totally self-checking operation is achieved through the proper partitioning of the product functions into two classes. This note presents a systematic method of such partitioning. Also, the note determines the cost of these checkers and compares them to previous designs.

The paper presents systematic procedures for decomposing a sequential machine into submachines some or all of which are realized by polylinear sequential circuits. This polylinear decomposition is based upon classes of subsets of the set of states which possess the substitution property rather than upon partitions with substitution property. These classes are easily found using the backward state transition function of the machine and enables one to treat uniformly and simultaneously the problems of state minimization, machine decomposition in the classical or the polylinear way, finding the lattice of the partitions with substitution property, and periodic decomposition based upon classes that correspond to cyclic partitions. Procedures are developed for deriving optimal all polylinear decompositions and "good" partial polylinear decompositions of a machine.