Some Unique Ternary Constant-composition Codes

The uniqueness of some ternary constant-composition codes is proved by combinato-rial methods. 1. Introduction. For all basic notions and facts about coding theory which are not introduced here we refer to [3]. All codes to be considered are ternary constant-composition codes. Ternary constant-composition (TCC) codes of length n are codes with constant composition of " zeros " , " ones " and " twos " and minimum distance d. Let (n 0 :n 1 :n 2 , M, d) code denote the TCC code with n 0 " zeros " , n 1 " ones " and n 2 " twos " in each codeword, M codewords and minimum distance d. Let A 3 (n 0 :n 1 :n 2 , d) denote the largest value of the size of the code M such that there exists an (n 0 :n 1 :n 2 , M, d) code. Codes with such parameters are called optimal. The fundamental question in coding theory is the existence of codes with given parameters. In cases when the existence problem has already been solved, the problem for the classification of all inequivalent codes with these parameters arises. The problem of finding values of A 3 (n 0 :n 1 :n 2 , d) is considered in [1], [4], [5], [2]. In this paper some TCC codes are enumerated up to equivalence. We obtain that some optimal TCC codes are unique. A family of unique TCC codes with parameters (t + λ:1:t − 1, (2t + λ)/t, 2t) for every integer t ≥ 2, λ ≥ 0 is found.