Perfect codes as exact minimizers of quadratic discrepancy in q-ary Hamming spaces

Abstract

Stolarsky's invariance principle converts quadratic discrepancy into an energy-minimization problem. Barg developed its form for binary Hamming space and proved that binary perfect codes minimize total quadratic ball discrepancy among binary codes of the same length and cardinality. We prove an exact finite-alphabet counterpart: whenever a fixed Hamming space and cardinality admit a perfect code, the perfect codes are precisely the discrepancy minimizers. The competitors are arbitrary subsets of the prescribed cardinality, with no linearity or minimum-distance assumption. More sharply, we give explicit parameter-only lower benchmarks before existence is known. For every arithmetically admissible one-error parameter set, and for every nontrivial two-error parameter set satisfying the sphere-packing and integral Lloyd-root conditions, equality in the corresponding benchmark is equivalent to perfect tiling. The bounds are proved by explicit Fourier--Krawtchouk and polynomial certificates rather than by solving a numerical linear program. Thus, in every covered parameter regime, perfect-code existence is equivalent to attainment of an explicit variational target.