A classical theorem on character degrees states that if a finite group has fewer than four degrees, then the is solvable. We prove corresponding result values by showing eight in its table, This confirms conjecture of T. Sakurai. also classify non-solvable groups with exactly values.

Syndrome decoding is a more efficient method of decoding linear codes over finite fields over a noisy channel [5]. Thus, in this paper we investigate the generalization of the syndrome decoding to linear codes over finite commutative rings. A first generalization was given in [1] via Pontryagin duality. In the same direction we give another generalization using linear functional-based duality. ...