Perfect nonbinary AN codes with distance three

It has been proved that the perfect binary and ternary single error correcting AN codes exist.In this paper, perfect radix-r single error correcting codes of type I and II are defined as generalized versions of perfect binary codes, and a general theory is developed for the existence of perfect single error correcting AN codes by using number theoretic concepts.This leads us to the followings:(1)There do not exist perfect radix 4k2 codes of type I and perfect radix k2 codes of type II.(2)In the cases of radix 4, 5, 8 and 9, no perfect code of either type exists.(3)There exist perfect codes with radices 6 and 7

Some aspects of linear space automata

Linear space automaton is introduced as a generalization of probabilistic automaton and its various properties are investigated.Linear space automaton has the abilities equivalent to probabilistic automaton but we can treat the former more easily than the latter because we can make use of properties of the linear space, successfully.First the solutions are given for the problems of connectivity, state equivalence, reduction and identification of linear space automata. Second, the matrix representation of linear space automaton is investigated and the relations between linear space automaton and probabilistic automaton are shown. Third, we discuss the closure properties of the family of all real functions on a free semigroup Σ* which are defined by linear space automata and then give a solution to the synthesis problem of linear space automata.Finally, some considerations are given to the problems of sets of tapes accepted by l.a.'s and also of operations under which the family of all the output functions of l.a.'s is not closed