The Spectrum Of (3,1,2)-r-conjugate Orthogonal Latin Squares

Latin square theory has been widely used in computer science, coding theory,secure communication and other fields, there are special concerns associated with theresearch on the spectrum of r-conjugate orthogonal Latin square. This study enablesus to obtain3pairs encryption-decryption algorithm from a quasigroup (Latin square)with proper encryption methods. When the order of a quasigroup reach to a certaindegree, we use the theory of stream cipher to encrypt the confidential information,then it is impossible to be decoded and confidentiality is very good. Therefore, itis of important significance for the study of the spectrum of r-conjugate orthogonalLatin square.The primary job for this paper is to study the spectrum of (3,1,2)-r-conjugateorthogonal Latin square and the main work is placed in the structure of a Latinsquare. Specific work is as follows:(1) We give the proof that there exists a (3,1,2)-r-COLS(v) for r∈{v, v~2} ap-plying TD construction.(2) We give the proof that there exists a (3,1,2)-r-COLS(v) for r∈{v~2-2, v~2-4, v~2-5, v~2-6, v~2-7} applying filling in holes construction.(3) For the order v <49, we mainly apply row permutation construction, fillingin holes construction and inflation constructions, at the same time, with the help ofcomputer search to obtain diferent r value of (3,1,2)-r-COLS(v).(4) For the order v≥49, we mainly apply GDD constructions and weightingconstruction with comprehensive use of various methods to obtain diferent r value of(3,1,2)-r-COLS(v). In particular, we have demonstrated that for any integer v≥162,the spectrum of a (3,1,2)-r-COLS(v) exists if and only if r∈{v, v+4, v+6}[v+8, v~2-4]{v~22, v~2}, except possibly for r∈{v+2, v+3, v+5, v+7, v~2-3}.(5) We give an application of Latin squares in cryptography.