In a certain code language, “CASTLE” is written as “BYPUNH”, and “DEM…

So,

C -1 B

A  -2 Y

S -3 P

T +1 U

L +2 N

E +3 H

And, 

D -1 C

E -2 C

M -3 J

A +1 B

N +2 P

D +3 G

Similarly,

E -1 D

I -2 G

T -3 Q

H +1 I

E +2 G

R +3 U

Thus, DGQIGU is correct.