F_q ~ N Are Permutation And Inversion Code

In 1995, Dr.L.Muttenthal in Teledyne Electric Technical Company (U.S.) proposed the orthomorphism permutation theory used for the code arithmetic and the design of confidential communication system. It was proved that the orthomorphism permutation has good character on cryptology.But the orthomorphism permutation theory was discussed in binary field. It is of interest to generalize the orthomorphism permutation on the F₂ⁿ to the one on the vector space F_qⁿ over the general finite field F_q and deduce the main properties of the orthomorphism permutation and maximal linear orthomorphism permutation on F_qⁿ. In addition,reversible codes are special cyclic codes. It is important in theory and practice to study reversible codes.In Chapter one of the thesis,we firstly review the definitions and properties of the orthomorphism permutation and maximal linear orthomorphism permutation on F₂ⁿ. Then, we generalize the orthomorphism permutation on the F₂ⁿ to the one on the vector space F_qⁿ over the general finite field F_q. After that we get three equivalent conditions of maximal linear orthomorphism permutation on F_qⁿ and a necessary and sufficient condition of making linear orthomorphism permutation maximal linear orthomorphism permutation on F_qⁿ Thereby we can work out numerous nonlinear orthomorphism permutations with better cryptological character.In Chapter two of the thesis, we introduce the reversible codes,and get main properties of reversible codes by the knowledge of codes and polynomials.