7 research outputs found
Compressed Neighbour Discovery using Sparse Kerdock Matrices
We study the network-wide neighbour discovery problem in wireless networks in
which each node in a network must discovery the network interface addresses
(NIAs) of its neighbours. We work within the rapid on-off division duplex
framework proposed by Guo and Zhang (2010) in which all nodes are assigned
different on-off signatures which allow them listen to the transmissions of
neighbouring nodes during their off slots, leading to a compressed sensing
problem at each node with a collapsed codebook determined by a given node's
transmission signature. We propose sparse Kerdock matrices as codebooks for the
neighbour discovery problem. These matrices share the same row space as certain
Delsarte-Goethals frames based upon Reed Muller codes, whilst at the same time
being extremely sparse. We present numerical experiments using two different
compressed sensing recovery algorithms, One Step Thresholding (OST) and
Normalised Iterative Hard Thresholding (NIHT). For both algorithms, a higher
proportion of neighbours are successfully identified using sparse Kerdock
matrices compared to codebooks based on Reed Muller codes with random erasures
as proposed by Zhang and Guo (2011). We argue that the improvement is due to
the better interference cancellation properties of sparse Kerdock matrices when
collapsed according to a given node's transmission signature. We show by
explicit calculation that the coherence of the collapsed codebooks resulting
from sparse Kerdock matrices remains near-optimal
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 over . We show that
exponentiating these -valued codewords by 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 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(). 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 -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
-designs on encoded qubits, i.e., to construct logical unitary -designs.
Software implementations are available at
https://github.com/nrenga/symplectic-arxiv18a, which we use to provide
empirical gate complexities for up to 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
Kerdock Codes Determine Unitary 2-Designs
The binary non-linear Kerdock codes are Gray images of β€_4-linear Kerdock codes of length N =2^m . We show that exponentiating Δ±=ββ-1 by these β€_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 β€_4-linear Kerdock codes of length N =2^m . We show that exponentiating Δ±=ββ-1 by these β€_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
Fundamental limits of many-user MAC with finite payloads and fading
Consider a (multiple-access) wireless communication system where users are
connected to a unique base station over a shared-spectrum radio links. Each
user has a fixed number of bits to send to the base station, and his signal
gets attenuated by a random channel gain (quasi-static fading). In this paper
we consider the many-user asymptotics of Chen-Chen-Guo'2017, where the number
of users grows linearly with the blocklength. In addition, we adopt a per-user
probability of error criterion of Polyanskiy'2017 (as opposed to classical
joint-error probability criterion). Under these two settings we derive bounds
on the optimal required energy-per-bit for reliable multi-access communication.
We confirm the curious behaviour (previously observed for non-fading MAC) of
the possibility of perfect multi-user interference cancellation for user
densities below a critical threshold. Further we demonstrate the suboptimality
of standard solutions such as orthogonalization (i.e., TDMA/FDMA) and treating
interference as noise (i.e. pseudo-random CDMA without multi-user detection).Comment: 38 pages, conference version accepted to IEEE ISIT 201