Fixed points of the self-power map over a finite field have been studied in cryptology as a special case of modular exponentiation. In this note, we define an elliptic-curve version of the self-power map, enumerate the number of curves that contain at least one fixed point, and give its upper and lower bounds. Our result is a partial solution to the open question raised by Glebsky and Shparlinski in 2010.
In this note, we give some restrictions on the number of vectors of weight d/2+1 in the shadow of a singly even self-dual [n,n/2,d] code. This eliminates some of the possible weight enumerators of singly even self-dual [n,n/2,d] codes for (n,d)=(62,12), (72,14), (82,16), (90,16) and (100,18).