A construction of 2-party card-based protocol over a Likert-type scale with 8 points and its relation to the association scheme H(2,3)

In ACNS2023 poster, we proposed a new card-based protocol between two parties using four cards, the two users could indicate how close their opinions are on a 4-point scale (0 = not at all disagree, 1 = somewhat disagree, 2 = somewhat agree, and 3 = strongly agree). Specifically, the protocol consists of three patterns: complete agreement, approximately agreement (cases with 0,1 or 2,3 input), and disagreement, where only the results are known while keeping the input secret from the third party. In the above discussions, we pointed out the relevance of the association schemes, and we report here a clarification of the relevance of the association schemes when we consider a natural extension to the case where each participant uses 3 up-down cards to input a secret over a Likert-type scale with 8 points.