## The Trifference Problem

Anurag Bishnoi wrote this beautiful post on a problem going back to a 1988 paper of Körner and Marton, and on a recent lovely paper by Anurag Bishnoi, Jozefien D’haeseleer, Dion Gijswijt, and Aditya Potukuchi, Blocking sets, minimal codes and trifferent codes.

What is the largest possible size of a set \$latex C\$ of ternary strings of length \$latex n\$, with the property that for any three distinct strings in \$latex C\$, there is a position where they all differ?

Let \$latex T(n)\$ denote this largest size. Trivially, \$latex T(1) = 3^1 = 3\$, and after some playing around you can perhaps prove that \$latex T(2) = 4\$ (I encourage you to try it so that you understand the problem). With a bit more effort, and perhaps the help of a computer, you might also be able to show that \$latex T(3) = 6\$, and \$latex T(4) = 9\$. For example, here is a set of nine ternary strings showing that \$latex T(4) geq 9\$: \$latex 0000, 0111, 2012, 2201, 2120, 0111, 1012, 1101, 1210\$. You should check that for any three strings from these nine, there is at least one position…

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### 2 Responses to The Trifference Problem

1. Thanks a lot Gil! Here is the arXiv preprint: https://arxiv.org/abs/2301.09457

GK: Thanks, Anurag. I updated my post…

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