Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

Unitals in projective planes of order 16

In this study, we perform computer searches for unitals in planes of order 16. The number of known nonisomorphic unitals in these planes is improved to be 261. Some data related to 2- (65, 5, 1) designs associated with unitals are given. New lower bounds on the number of unital designs in projective planes of order 16 and 2- (65, 5, 1) designs are established. The computations show that thirty-nine unitals can be embedded in two or more nonisomorphic projective planes of order 16. Fifteen new connections between planes of order 16 (based on unitals) are found. All unitals found by the algorithms used in this study are explicitly listed. We assume familiarity with the basic facts from combinatorial design theory and finite geometries [5, 9, 16]. A t-(v, k, ?) design (t-design) is a pair D = {X, B} of a set X of cardinality v, called points, and a collection B of k-subsets of X, called blocks, such that every t points appear together in exactly ? blocks. A 2-design with ? = 1 is called a Steiner design. The incidence matrix of a 2-(v, k, ?) design D is a matrix M = (mij) with rows labeled by the blocks of D, columns labeled by the points of D, where mi,j = 1 if the ith block contains the j th point and mi,j = 0 otherwise. For a prime p, the rank of the incidence matrix of design D over a finite field of characteristic p is called the p-rank of D. Two designs D and D? are called isomorphic if there is a bijection between their point sets that map

The Desarguesian projective plane of order eleven and related codes

In a finite projective plane PG(2; q), a set K of k points is a (k; n)-arc for 2 ≤ n ≤ q - 1 if the following two properties hold: 1. Every line intersects K in at most n points. 2. There exists a line which intersects K in exactly n points. Algebraic curves of degree n give examples of (k; n)-arc; the parameter n is called the degree of the arc. In PG(2; q); the problem of finding mn(2; q) and tn(2; q) (the maximum and the minimum value of k for which a complete (k; n)-arc exists) and the problem of classifying such arcs up to projective equivalence, are crucial problems in finite geometry. One of the important application of these arcs in coding theory are projective codes that cannot be extended to larger codes. The aim of this project is to classify (k; n)-arcs if possible for 3 ≤ n ≤ 5 and to construct large arcs in PG(2; 11): Algebraic and new combinatorial methods are used to perform the classification and the construction of such arcs with different degrees. Those procedures are implemented using different open-source software packages such as GAP [35] and Orbiter [10]. We were successful in obtaining new isomorphism types of (k; 5)-arcs for k = 5,…, 13 in PG(2; 11): We have also developed a new classification algorithm for cubic curves in small projective planes. Moreover, a new upper bound is proved for the number of 5-secants of (45; 5)-arc. In addition to proving our new lower bound for the complete (k; 5)-arc in PG(2; 11): The non existence of (44; 5)-arc and (45; 5)-arc is formulated as a new conjecture for q = 11: Using an arc of degree 2 and exploiting the complement relation between arcs and blocking sets we find new 134 isomorphism types of (77; 8)-arcs in PG(2; 11)