Twisted Fourier Analysis of Quantum Weight Enumerators
Ching-Yi Lai
A unified Fourier framework for quantum weight enumerators of prime-power-dimensional qudits is developed in this paper. We introduce the notions of Weyl power spectrum and Weyl-conjugation autocorrelation for the Weyl-operator quantities nderlying the Shor--Laflamme weight enumerators. Representing these quantities as formal vectors, we show that they form a symplectic Fourier transform pair, from which the quantum MacWilliams identity follows by applying the Hamming-weight functional. This framework extends to a family of twisted Fourier transforms, including the Rains shadow transform. We show that identical-copy Bell sampling corresponds to a twisted autocorrelation related to the shadow enumerator through Weyl translation and Hamming projection. We further prove that all twisted Fourier transforms are Weyl translations of either the ordinary MacWilliams transform or the Rains shadow transform. This two-class reduction implies that the twists yield no additional independent coefficientwise positivity constraints under compatible projections. Nevertheless, the framework leads to refined MacWilliams and shadow identities that retain more Fourier information than ordinary Hamming weight enumerators. We employ these refined identities in linear programming, together with simplex mapping and shadow-coset counting, to exclude surviving candidate weight distributions. These techniques yield tighter bounds for binary and qutrit quantum codes, including nonexistence results for specific code parameters.
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