Byte error control codes for data transmission and arithmetic units

Advances in the speed and complexity of modern digital systems where the integration level moves toward multi-million transistor integrated circuits, the need arises for error control codes with greater capability than those used in the past. Error correcting codes, even those which require large amounts of hardware, can now be embedded within the integrated circuit itself. Codes that in the past were used for random error correction can now be reconsidered for multi-bit error detection and correction.; Arithmetic operations require codes that differ from the communication codes. Therefore one of the goals of this research is to develop error detecting and/or correcting codes of dual use for both the arithmetic and transmission units. AN codes, particularly the large distance codes are formally classified, their distance properties evaluated, and the feasibility of their use as dual codes considered. A breakthrough in coding theory established that certain transmission codes, such as the non-linear Nordstrom-Robinson code, are linear over the ring of integers modulo 4 (Z₄). Since the non-linear Nordstrom-Robinson code can be handled as linear when mapped into Z₄ it is reexamined here to determine its ability to also handle byte and multiple errors.; A class of arithmetic codes, called AN codes, are investigated and the condition for a code to be cyclic is broadened to include negacyclic codes. An expression for the arithmetic distance of a class of AN codes with composite A composed of two and only two prime integers is presented. It is shown that this composite A must have a restricted set of primitive roots and a proof is provided for the class of allowable values of A. This class of AN codes is shown to be a generalized form of the Barrows-Mandelbaum codes and a number of examples are provided.; High-speed multiplication carried out by dividing the multiplier into r m-bit blocks, where each block multiplies the multiplicand forming a partial product. An error, that occurs because of faulty multiplier circuits, is repeated up to r times and called an iterative error. With the proper choice of check base A, these iterative errors are able to be detected. This is proved here for m − 1 errors where m ≥ 2.; The Nordstrom-Robinson code is shown to detect and correct multiple errors if the errors are restricted to a single m-bit byte. The Generator and Parity check matrices are derived and examples of Z ₄ linearity are given. A constructive method of evaluating linear codes using Z₄ linearity is demonstrated by comparing with linear methods from the literature. Using Z ₄ linearity, it is shown that the Nordstrom-Robinson code is capable of correcting 3-bit errors occurring in one 8-bit byte.