On relations between error correcting codes and multi-valued logic functions via discrete Fourier transforms

2021 Fiscal Year Final Research Report

• Project/Area Number: 19K22850
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Review Section: Medium-sized Section 60:Information science, computer engineering, and related fields
• 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): 2019-06-28 – 2022-03-31
• Keywords: 準巡回符号 / 自己双対符号 / 自己直交符号 / 反転不変符号 / 巡回符号 / 最小重み / 有限体 / 中国剰余定理

Outline of Final Research Achievements

1. Generator polynomial matrices of quasi-cyclic (QC) codes obtained from cyclic codes over extended finite fields have been determined. For a QC code Q with the generator polynomial matrix G, a necessary and sufficient condition for G which corresponds to a QC code obtained from a cyclic code over the extended finite field has been presented. As their application, the spectrums of cyclic codes over extended finite fields which produce reversible QC codes have been decided.
2. We conducted research on general QC codes and determined the conditions of generator polynomial matrices for reversible, self-orthogonal, and self-dual QC codes. Through computer search with results of this study, various reversible self-orthogonal QC codes whose minimum distances achieve their upper bounds of have been found.

Free Research Field

情報理論

Academic Significance and Societal Importance of the Research Achievements

準巡回符号と呼ばれる誤り訂正符号のクラスについて，生成多項式行列を軸とした研究を行った．これまで研究代表者は，双対符号に対する生成多項式行列の公式を求め，自己直交および自己双対符号の構成と探索に応用してきた．本研究では，反転符号に対する生成多項式行列の公式を求め，反転不変符号の構成と探索に応用した．また，素因子分解および中国剰余定理を応用することにより，自己直交符号および反転不変符号の構成と探索が高速化されることが判明したため，この手法によりこれらのクラスの誤り訂正能力の高い符号をリストアップした．