As a new extension of a cubic set, the notion of a cubic interval-valued intuitionistic fuzzy set is introduced, and its application in $BCK/BCI$ -algebra is considered. The notions of $\alpha$ -internal, $\beta$ -internal, $\alpha$ -external and $\beta$ -external cubic IVIF set are introduced, and the P-union, P-intersection, R-union and R-intersection of $\alpha$ -internal and $\alpha$ -external cubic IVIF sets are discussed. The concepts of cubic IVIF subalgebra and ideal in $BCK/BCI$ -algebra are introduced, and related properties are investigated. Relations between cubic IVIF subalgebra and cubic IVIF ideal are considered, and characterizations of cubic IVIF subalgebra and cubic IVIF ideal are discussed.

For $i,j,k,l,m,n\in \{1,2,3,4\}$ , the notion of $\left(T\right(i,j),I(k,l),F(m,n\left)\right)$ -interval neutrosophic subalgebra in $BCK/BCI$ -algebra is introduced, and their properties and relations are investigated. The notion of interval neutrosophic length of an interval neutrosophic set is also introduced, and related properties are investigated.

The notions of hyperfuzzy ideals in $BCK/BCI$ -algebras are introduced, and related properties are investigated. Characterizations of hyperfuzzy ideals are established. Relations between hyperfuzzy ideals and hyperfuzzy subalgebras are discussed. Conditions for hyperfuzzy subalgebras to be hyperfuzzy ideals are provided.

As a generalization of interval-valued fuzzy sets and fuzzy sets, the concept of hyperfuzzy sets was introduced by Ghosh and Samanta in the paper [J. Ghosh and T.K. Samanta, Hyperfuzzy sets and hyperfuzzy group, Int. J. Advanced Sci Tech. 41 (2012), 27–37]. The aim of this manuscript is to introduce the length-fuzzy set and apply it to $BCK/BCI$ -algebras. The notion of length-fuzzy subalgebras in $BCK/BCI$ -algebras is introduced, and related properties are investigated. Characterizations of a length-fuzzy subalgebra are discussed. Relations between length-fuzzy subalgebras and hyperfuzzy subalgebras are established.

It is known that $42$ is the largest size of a $6$-arc in the Desarguesian projective plane of order $8$. In this paper, we classify these $(42,6)_8$ arcs. Equivalently, we classify the smallest $3$-fold blocking sets in PG$(2,8)$, which have size $31$.