Construction of high-performance error-correcting codes using Grobner bases

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13994
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Communication/Network engineering
• Research Institution: Toyota Technological Institute
• Principal Investigator: Matsui Hajime 豊田工業大学, 工学(系)研究科(研究院), 准教授 (80329854)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 離散フーリエ変換 / 多値論理関数 / 準巡回符号 / 離散Fourier変換 / ユークリッド整域 / 畳み込み定理 / 整数符号 / 代数的符号

Outline of Final Research Achievements

1. A fast decoding method of projective Reed-Muller codes has been established (collaboration with Dr. Norihiro Nakashima). Compared to decoding using the Gaussian elimination method, the order of its computational complexity could be reduced.
2. The error-correcting codes over the residue rings of the Euclidean domains have been investigated. They include quasi-cyclic codes and integer codes as special cases. We showed that we can uniquely determine the generator matrix for each code. This fact is useful for code construction and search.
3. The convolution theorem for a class of multiple-valued logic polynomials has been established and a fast calculation method for the multiplication of them has been derived. We showed that the discrete Fourier transform used in the convolution theorem has a transposition relation to that used for decoding affine variaty codes.

Free Research Field

情報理論