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

Higher-order CIS codes

We introduce {\bf complementary information set codes} of higher-order. A binary linear code of length $tk$ and dimension $k$ is called a complementary information set code of order $t$ ($t$-CIS code for short) if it has $t$ pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a $t$-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order $3$ is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length $\le 12$ is given. Nonlinear codes better than linear codes are derived by taking binary images of $\Z_4$-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate $1/t$ it either provides $t$ disjoint information sets or proves that the code is not $t$-CIS. Using this algorithm, all optimal or best known $[tk, k]$ codes where $t=3, 4, \dots, 256$ and $1 \le k \le \lfloor 256/t \rfloor$ are shown to be $t$-CIS for all such $k$ and $t$, except for $t=3$ with $k=44$ and $t=4$ with $k=37$.Comment: 13 pages; 1 figur

Error Correction for Cooperative Data Exchange

This paper considers the problem of error correction for a cooperative data exchange (CDE) system, where some clients are compromised or failed and send false messages. Assuming each client possesses a subset of the total messages, we analyze the error correction capability when every client is allowed to broadcast only one linearly-coded message. Our error correction capability bound determines the maximum number of clients that can be compromised or failed without jeopardizing the final decoding solution at each client. We show that deterministic, feasible linear codes exist that can achieve the derived bound. We also evaluate random linear codes, where the coding coefficients are drawn randomly, and then develop the probability for a client to withstand a certain number of compromised or failed peers and successfully deduce the complete message for any network size and any initial message distributions