Kerdock Codes Determine Unitary 2-Designs

Abstract

The binary non-linear Kerdock codes are Gray images of ℤ_4-linear Kerdock codes of length N =2^m . We show that exponentiating ı=−√-1 by these ℤ_4-valued codewords produces stabilizer states, which are the common eigenvectors of maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the proof of the classical weight distribution of Kerdock codes. Next, we partition stabilizer states into N +1 mutually unbiased bases and prove that automorphisms of the Kerdock code permute the associated MCS. This automorphism group, represented as symplectic matrices, is isomorphic to the projective special linear group PSL(2,N) and forms a unitary 2-design. The design described here was originally discovered by Cleve et al. (2016), but the connection to classical codes is new. This significantly simplifies the description of the design and its translation to circuits.