Sergey A. Inutin
Surgut State University, Russia
emails: lyubov@iastate.edu and inyutin@surguttel.ru
Controlling the results of computations is often the only way to evaluate the reliability of the results of time-consuming, high algorithmic complexity computational processes. To guarantee the correctness of the computation results when the computational variables vary in super large diapasons, it is necessary to verify the correctness of the results of the intermediate steps of the computational process. A noise-resistant modular arithmetic code may ensure complete control of the computational process or the process' components and/or steps that bear the major volume and time of computations.
The paper proposes a parallel noise-resistant modular code of a quadratic diapason and investigates the relationship between metrics in the linear space of the vectors of modular components. Also, a method and algorithm of syndrome decoding combined with the computation of the positional functional is investigated.
It is shown that the introduced modular code has the following properties:
1. It is an algebraic ring and is capable of controlling all ring operations and any operations based on ring operations.
2. Regular, informational, and control components are being processed by the same arithmetic algorithms of the computational process. The coding and decoding procedures cannot be fully reassembled because of the large number of the bases of the modular system.
3. The algorithms of coding and decoding are syndrome; they are combined with the algorithms of computation of the positional functionals used to execute non-modular operations.
4. The code is linear.
5. The code's informational and control components are kept separate.
6. The condition of dense packing is not satisfied for the code.
7. The modular number system of the quadratic diapason is a modular noise-resistant non-surplus code.