In a finite projective plane \( \text{PG}(2, q) \), a set of \( k \) points is called a \( (k, n) \)-arc if the following two properties hold:
1. Every line intersects it in at most \( n \) points.
2. There exists a line which intersects it in exactly \( n \) points.

We are interested in determining, for each \( q \) and each \( n \), the largest value of \( k \) for which a \( (k, n) \)-arc exists in \( \text{PG}(2, q) \). If possible, we would like to classify those arcs up to isomorphism. We look at the problem for \( q = 11 \).

A cyclic triple, \( (a, b, c) \), is defined to be the set \( \{(a, b), (b, c), (c, a)\} \) of ordered pairs. A Mendelsohn triple system of order \( v \), or MTS\( (v) \), is a pair \( (M, \beta) \), where \( M \) is a set of \( v \) points and \( \beta \) is a collection of cyclic triples, each containing pairwise distinct points of \( M \) such that every ordered pair of distinct points of \( M \) exists in exactly one cyclic triple of \( \beta \). An antiautomorphism of a Mendelsohn triple system \( (M, \beta) \) is a permutation of \( M \) which maps \( \beta \) to \( \beta^{-1} \), where \( \beta^{-1} = \{(c, b, a) \mid (a, b, c) \in \beta\} \). Necessary conditions for the existence of an MTS\( (v) \) admitting an antiautomorphism consisting of two cycles of lengths \( M \) and \( N \), where \( 1 < M \leq N \), have been shown, and for the cases of \( N = M \) and \( N = 2M \), sufficiency has been shown. We show sufficiency for the cases in which \( M = 13 \) and \( N = 78, 390, \) and \( 702 \).

The study of the generalized Fermat variety

\[
\phi_j = \frac{x^j + y^j + z^j + (x+y+z)^j}{(x+y)(x+z)(y+z)}
\]

defined over a finite field \( L = \mathbb{F}_q \), where \( q = 2^n \) for some positive integer \( n \), plays an important role in the study of (APN) functions and exceptional APN functions. This study arose after a characterization by Rodier that relates these functions with the number of rational points of \( \phi_j = (x,y,z) \). The most studied cases are when \( j = 2^k + 1 \) and \( j = 2^{2k} – 2^{k} + 1 \), the Gold and Kasami-Welch numbers. In this article, we make a claim about the decomposition of \( \phi_j \) into absolutely irreducible components. If these components intersect transversally at a particular point, then the corresponding Kasami-Welch polynomial is absolutely irreducible. This implies that the function is not exceptional APN, thus helping us make progress on the stated conjecture.

For \( n \geq 1 \), let \( a_n \) count the number of strings \( s_1 s_2 s_3 \ldots s_n \), where
(i) \( s_1 = 0 \);
(ii) \( s_i \in \{0, 1, 2\} \) for \( 2 \leq i \leq n \);
(iii) \( |s_i – s_{i-1}| \leq 1 \) for \( 2 \leq i \leq n \).

Then \( a_1 = 1 \), \( a_2 = 2 \), \( a_3 = 5 \), \( a_4 = 12 \), and \( a_5 = 29 \).

In general, for \( n \geq 3 \), \( a_n = 2a_{n-1} + a_{n-2} \), and \( a_n \) equals \( P_n \), the \( n \)th \emph{Pell} number.

For these \( P_n \) strings of length \( n \), we count
(i) The number of occurrences of each symbol \( 0, 1, 2 \);
(ii) The number of times each symbol \( 0, 1, 2 \) occurs in an even or odd position;
(iii) The number of levels, rises, and descents within the strings;
(iv) The number of runs that occur within the strings;
(v) The sum of all strings considered as base \( 3 \) integers;
(vi) The number of inversions and coinversions within the strings; and
(vii) The sum of the major indices for the strings.

A family of graphs, called Generalized Johnson graphs, provides an abstraction of both Kneser and Johnson graphs.
Given the symmetric nature of Generalized Johnson graphs, we provide various decompositions of these graphs and demonstrate non-trivial instances of the impossibility of decomposing such graphs into triples.