OPTIMAL ERROR CORRECTING CODES AND THEIR APPLICATIONS. SOFTWARE ON CODING THEORY

THIS PROJECT REPRESENTS A CONTINUATION OF FIVE PREVIOUS PROJECTS: THE PROJECT ENTITLED "OPTIMAL CODES. NEW CONSTRUCTIONS AND APPLICATIONS" (MTM2009-08435) WHERE THE MAIN GOAL WAS TO STUDY OPTIMAL COMBINATORIAL CODES AND SOME APPLICATIONS TO CRYPTOGRAPHY, SPECIFICALLY TO STEGANOGRAPHY; THE PROJECT ENTITLED "INFORMATION CODING FOR DISTRIBUTED STORAGE AND AUTHENTICATION IN SOCIAL NETWORKS" (TIN2010-17358) WHERE THE MAIN OBJECTIVE WAS THE APPLICATION OF OPTIMAL CODES AND NETWORK CODING IN THE FIELD OF DISTRIBUTED STORAGE AND INTEGRITY OF STORED DATA; THE PROJECT ENTITLED "OPTIMAL CODES AND THEIR APPLICATIONS TO CRYPTOGRAPHY AND DISTRIBUTED STORAGE. SOFTWARE FOR EXPERIMENTATION IN CODING THEORY" (TIN2013-40524-P) WHOSE PURPOSE WAS TO CONTINUE THE ADVANCEMENT OF KNOWLEDGE ON OPTIMAL REGULAR CODES AND ITS APPLICATIONS IN THE TWO AFOREMENTIONED FIELDS, AND TO CONTINUE THE DEVELOPMENT OF SOFTWARE TOOLS IN MAGMA FOR EXPERIMENTATION AND SIMULATION OF ERROR CORRECTING CODES BEHAVIOR; THE PROJECT ENTITLED "ERROR CORRECTING CODES AND THEIR APPLICATIONS: COMPLETELY REGULAR AND HADAMARD. SOFTWARE IN CODING THEORY" (TIN2016-77918-P) WHERE PROGRESS WAS MADE IN THE STUDY OF OPTIMAL ERROR-CORRECTING CODES, ESPECIALLY COMPLETELY REGULAR AND HADAMARD CODES, AND NEW FUNCTIONS FOR THE LIBRARY IN MAGMA ON Q-ARY CODES NOT NECESSARILY LINEAR WERE IMPLEMENTED; AND THE PROJECT ENTITLED "ERROR-CORRECTING CODES AND THEIR APPLICATIONS: DISTRIBUTED STORAGE AND QUANTUM COMPUTING. SOFTWARE IN CODING THEORY" (PID2019-104664GB-I00) WHERE REGULAR AND LOCALLY REPAIRABLE CODES APPLIED TO DISTRIBUTED STORAGE; HADAMARD ADDITIVE CODES AND THEIR APPLICATION TO QUANTUM COMPUTING; AND CODES OVER RINGS Z/P^S WITH P PRIME ARE STUDIED. THE MAGMA FUNCTIONS FOR THE PACKAGE ON Q-ARY CODES ARE ALSO EXTENDED AND THE CREATION OF A NEW PACKAGE FOR CODES OVER RINGS Z/P^S IS STARTED. TAKING INTO ACCOUNT THE RESULTS FROM THE PREVIOUS PROJECTS, IN THIS PROPOSAL, WE INTEND TO: 1) CONSTRUCT AND CLASSIFY COMPLETELY REGULAR CODES AND BENT FUNCTIONS, WITH APPLICATION TO CRYPTOGRAPHY. 2) ADVANCE IN THE STUDY AND CLASSIFICATION OF ADDITIVE CODES (HADAMARD, SIMPLEX, MACDONALD) OVER MIXED ALPHABETS, DIFFERENT COPIES OF THE RINGS Z/P^S, AND THE PERMUTATION DECODING METHOD FOR SUCH CODES. 3) CLASSIFY FAMILIES OF ADDITIVE CODES OVER FINITE FIELDS WITH APPLICATION TO QUANTUM COMPUTING. 4) CONTINUE WITH THE DEVELOPMENT OF THE MAGMA SYSTEM FOR SIMULATION AND EXPERIMENTATION IN CODING THEORY, EXTENDING THE FUNCTIONS OF THE PACKAGE FOR NON-LINEAR Q-ARY CODES OVER FINITE FIELDS AND THE PACKAGE FOR CODES OVER RINGS Z/P^S. BECAUSE OF THE EXPERIENCE IN THE PREVIOUS PROJECTS, AND THE NUMEROUS PUBLICATIONS OF THE MEMBERS OF OUR GROUP, AND OTHER INTERNATIONAL GROUPS, IT CAN BE CONCLUDED THAT ERROR CORRECTING CODES ARE AN EFFECTIVE TOOL IN MANY DIFFERENT ENVIRONMENTS WHERE THERE IS A PROCESS OF TRANSMISSION OF INFORMATION. ACCORDING TO THE OBJECTIVES AND EXPECTED RESULTS, WE INTEND TO PUBLISH ALL RESULTS IN HIGH IMPACT SCIENTIFIC JOURNALS, AS HAS BEEN DONE IN THE PREVIOUS PROJECTS GRANTED.