You can do better. X86 has both "op [mem], reg" and "op reg, [mem]" variants of most instructions, where "[mem]" can be a register too. So you have two ways to encode "xor eax, eax", differing by which of the operands is in the "possible memory operand" slot, the source or the destination.

This one would be a fun challenge in a ctf, or maybe more appropriate for a puzzle hunt – most people would look at the dissassembly and not at the actual bytes and completely miss the binary encoding