Bounds on Reversible, Complement, Reversible-Complement, Constant Weight Sum Codes

For an even-size alphabet, a code is called Reversible, Complement, Reversible-Complement, Constant Weight Sum code (RCCWS code) if the code is closed under reversible, complement, and reversible-complement sequences, and for some given symbols in the alphabet, the total number of occurrence of those symbols in each sequence is identical. In this work, we have obtained an upper bound (Sphere Packing bound) and a lower bound (Gilbert-Varshamov bound) on the maximum size of an RCCWS code defined over the even-size alphabet. As a special case of these two bounds, we also have obtained an upper and lower bound on the maximum size of DNA codes that satisfies reversible, complement, reversible-complement, and $GC$-content constraints.