The maximum cardinality of trifferent codes with lengths 5 and 6

Trifferent Codes with Small Lengths

“…In Table 1 we have listed the number a(n, l) of non-equivalent trifferent codes C with length n and cardinality l for all n ≤ 6. Clearly, we have a(n, 1) = 1 and a(n, 2) = n -counting the number of possibilities of the Hamming distance 1 The count a(6, 13) = 3 was also obtained in [DFGP22].…”

“…One way to represent trifferent codes is to write the codewords as columns of a matrix. As an example we state the three trifferent codes of length n = 6 and cardinality T (6) = 13 from [DFGP22]:   0 0 2 2 2 2 2 0 1 1 1 1 1 0 1 0 2 2 2 1 2 0 1 1 1 2 0 1 1 0 2 1 2 2 2 0 1 2 1 0 1 1 1 0 2 2 2 2 2 0 1 1 0 1 1 2 1 0 2 2 2 1 2 0 1 0 1 2 1 1 1 0 2 1 2 2 2 0   ,   0 0 2 2 2 2 2 0 1 1 1 1 1 0 1 0 2 2 1 2 2 0 1 1 2 1 0 1 1 0 2 2 1 2 2 0 1 1 2 0 1 1 1 0 2 2 2 2 2 0 1 1 0 1 2 1 1 0 2 2 1 2 2 0 1 0 1 1 2 1 1 0 2 2 1 2 2 0   ,…”

Blocking sets, minimal codes and trifferent codes

“…Let 𝑇(𝑛) denote the largest size of a trifferent code of length 𝑛. The exact value of 𝑇(𝑛) is only known for 𝑛 up to 10, where the last six values were obtained very recently via computer searches [28,41]. Asymptotically, the upper bound…”

“…Let $$T(n)$$ denote the largest size of a trifferent code of length $$n$$. The exact value of $$T(n)$$ is only known for $$n$$ up to 10, where the last six values were obtained very recently via computer searches [28, 41]. Asymptotically, the upper bound $$$$\begin{equation} T(n) \leqslant 2 {(3/2)}^n \end{equation}$$$$obtained by Körner [39] in 1973 is still the best known upper bound, despite considerable effort (see, for example, [23] where it is shown that a direct application of the slice rank method will not improve the bound.)…”

Blocking sets, minimal codes and trifferent codes

“…We will focus on the q = 3 case where these codes are also known as trifferent codes, and the problem of determining their largest possible size is called the trifference problem. Let T (n) denote the largest size of a trifferent code of length n. The exact value of T (n) is only known for n up-to 6, where the last two values were obtained very recently via computer searches [24]. Asymptotically, the following result of Körner [35] from 1973 is still the best known upper bound, despite considerable effort (see for example [19] where it is shown that a direct application of the slice rank method will not improve the bound)…”