Abstract
This paper presents and first scientifically substantiates the generalized theory of binomial number systems (BNS) and the method of their formation for reliable digital signal processing (DSP), transmission, and data storage. The method is obtained based on the general theory of positional number systems (PNS) with conditions and number functions for converting BNS with a binary alphabet, also allowing to generate matrix BNS, linear-cyclic, and multivalued number systems. Generated by BNS, binomial numbers possess the error detection property. A characteristic property of binomial numbers is the ability, on their basis, to form various combinatorial configurations based on the binomial coefficients, e.g., compositions or constant-weight (CW) codes. The theory of positional binary BNS construction and generation of binary binomial numbers are proposed. The basic properties and possible areas of application of BNS researched, particularly for the formation and numbering of combinatorial objects, are indicated. The CW binomial code is designed based on binary binomial numbers with variable code lengths. BNS is efficiently used to develop error detection digital devices and has the property of compressing information.
Highlights
Positional number systems (PNS) solve two main tasks, namely determining the amount and arranging information about different objects
The binomial numbers generated by the binomial number systems (BNS) divide into the first and second classes, characterized by 0 or 1 in the LSB of the binomial code
Each class is divided into groups with binomial codes of the same code length within the groups and different code lengths between the groups, differing by one bit
Summary
Positional number systems (PNS) solve two main tasks, namely determining the amount and arranging information about different objects. The PNS uses numeric numbering according to specific rules and provides arithmetic operations’ performance on them. Conditions of the PNS form the numbers used by number systems, determine the rules for performing arithmetic operations on them, and, if adding redundancy, make it possible to detect errors. Several classes of BNS are being developed These are binary, multivalued, linear-cyclic, matrix systems [20,21]. BNS can generate and number based on its digits various combinatorial configurations, which makes its use preferable to other number systems when solving several information tasks and, especially, in the circuit implementation of the corresponding digital devices [20,21].
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