Two Methods for Constructing Minimal (b,t)-Blocking Sets in PG(2,5) with Associated Complete Arcs and Projective Codes

الملخص

Minimal (b,t)-blocking sets constitute a significant class of combinatorial configurations in finite projective geometry because of their close connections with complete arcs and projective linear codes. This study presents a systematic construction and comparison of minimal (b,t)-blocking sets in the projective plane PG(2,5), for 1 ≤ t ≤ 5, using two distinct approaches: the Rédei-type method and the conic-type method. For each construction, both the blocking property and minimality of the resulting sets are verified. The Rédei-type method produces blocking sets of cardinalities 10,15,20,25 and 30, whereas the conic-type method yields sets of cardinalities 9,16,20,25 and 30, corresponding respectively to t=1,2,3,4 and 5. For n=5,4,3 and 2, respectively, the Rédei-type construction generates complete (k,n)-arcs of sizes k=21,16,11 and 6. These arcs give rise to the projective linear codes , , , and , respectively. Similarly, the conic-type construction produces complete (k,n)-arcs of sizes k=22,15,11 and 6, corresponding respectively to n=5,4,3 and 2, from which the projective linear codes , , , and are derived. The obtained results demonstrate the effectiveness of both construction methods and establish a clear structural connection among minimal blocking sets, complete arcs, and projective linear codes in PG(2,5). Future studies may extend this investigation to minimal (b,t)-blocking sets in PG(2,5) for q=7,9,11,13, as well as for other prime-power values, together with the complete arcs and projective linear codes arising from these configurations.