18 research outputs found
New characterisations of the Nordstrom–Robinson codes
In his doctoral thesis, Snover proved that any binary code
is equivalent to the Nordstrom-Robinson code or the punctured
Nordstrom-Robinson code for or respectively. We
prove that these codes are also characterised as \emph{completely regular}
binary codes with or , and moreover, that they are
\emph{completely transitive}. Also, it is known that completely transitive
codes are necessarily completely regular, but whether the converse holds has up
to now been an open question. We answer this by proving that certain completely
regular codes are not completely transitive, namely, the (Punctured) Preparata
codes other than the (Punctured) Nordstrom-Robinson code
Kerdock Codes Determine Unitary 2-Designs
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
Kerdock Codes Determine Unitary 2-Designs
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