Algorithmic Study of the Pascal-Type Triangle and Error-Correcting Codes

The structure of a Pascal-type triangle in a positional numeral system with base p is studied. Recursive functions in the object-oriented programming language C++ are used to derive the structure of a Pascal-type triangle modulo p. A formula for the trinomial coeficient, obtained using two binomial coeficients, is used in the study. An algorithm is developed for identifying the properties of working code combinations from Pascal-type tables used for errorcorrecting encoding and decoding of information. A boundary condition relating the number of encoded messages, the positional system, and the length of the encoded message is considered. The maximum number of errors that can be corrected is identified. The program implements code redundancy and Hamming distance formulas. Possible working combinations obtained based on the Hamming distance are analyzed.