Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv packag

Coding Theory and Algebraic Combinatorics

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in Information and Coding Theory", ed. by I. Woungang et al., World Scientific, Singapore, 201

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.Comment: 16 page

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

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page

Automorphism groups and new constructions of maximum additive rank metric codes with restrictions

Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank of their difference is greater than or equal to $d$. A $d$-code with as many as possible elements is called a maximum $d$-code. The integer $d$ is also called the minimum distance of the code. When $d<n$, a classical example of such an object is the so-called generalized Gabidulin code. There exist several classes of maximum $d$-codes made up respectively of symmetric, alternating and hermitian matrices. In this article we focus on such examples. Precisely, we determine their automorphism groups and solve the equivalence issue for them. Finally, we exhibit a maximum symmetric $2$-code which is not equivalent to the one with same parameters known so far