Enumeration of subsets with closedness in finite fields of characteristic 2

Résumé

The additive closedness in the subset of an additive group is termed as $r$-value. The nature of closedness in different subsets of fixed size is observed as a spectrum of $r$-values. We enumerate $r$-values of subsets in finite fields of characteristic 2 and represent them as the spectrum of values. Based on these values the subsets can be further studied as partial Steiner triple systems, sum-free sets, Sidon sets, and Schure triples.