Cyclotomic Trace Codes

A generalization of Ding’s construction is proposed that employs as a defining set the collection of the sth powers ( s ≥ 2 ) of all nonzero elements in G F ( p m ) , where p ≥ 2 is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over G F ( 4 ) that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs.

Partitions on the Projective Line and Plane of Order 17k,k=1,2,3

In this research, the main purposes are making partitions of the projective line, PG (1, q3), q=17, and embedding the projective line, PG (1, q ) into PG (1, q3), as well as finding some partitions of PG (2, q2) as subplanes, orbits, triangles and arcs, and studying the properties of these subsets. Furthermore, the arcs with different degrees and sizes are found besides the embedding of the projective plane, PG (2,q) into PG (2, q2). In the coding theory, there are 14 projective codes that are introduced.

The construction of Complete (kn,n)-arcs in The Projective Plane PG(2,5) by Geometric Method, with the Related Blocking Sets and Projective Codes

A (k,n)-arc is a set of k points of PG(2,q) for some n, but not n + 1 of them, are collinear. A (k,n)-arc is complete if it is not contained in a (k + 1,n)-arc. In this paper we construct complete (kn,n)-arcs in PG(2,5), n = 2,3,4,5, by geometric method, with the related blocking sets and projective codes.