Related papers: Constructive spherical codes on layers of flat tor…
We present a new systematic approach to constructing spherical codes in dimensions $2^k$, based on Hopf foliations. Using the fact that a sphere $S^{2n-1}$ is foliated by manifolds $S_{\cos\eta}^{n-1} \times S_{\sin\eta}^{n-1}$,…
A new design method for high rate, fully diverse ('spherical') space frequency codes for MIMO-OFDM systems is proposed, which works for arbitrary numbers of antennas and subcarriers. The construction exploits a differential geometric…
The surface code is a two-dimensional topological code with code parameters that scale optimally with the number of physical qubits, under the constraint of two-dimensional locality. In three spatial dimensions an analogous simple yet…
We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat…
Shannon gave a lower bound in 1959 on the binary rate of spherical codes of given minimum Euclidean distance $\rho$. Using nonconstructive codes over a finite alphabet, we give a lower bound that is weaker but very close for small values of…
Long linear codes constructed from toric varieties over finite fields, their multiplicative structure and decoding. The main theme is the inherent multiplicative structure on toric codes. The multiplicative structure allows for…
We introduce tile codes, a simple yet powerful way of constructing quantum codes that are local on a planar 2D-lattice. Tile codes generalize the usual surface code by allowing for a bit more flexibility in terms of locality and stabilizer…
We present constructions of Space-Time (ST) codes based on lattice coset coding. First, we focus on ST code constructions for the short block-length case, i.e., when the block-length is equal to or slightly larger than the number of…
A new class of space time codes with high performance is presented. The code design utilizes tailor-made permutation codes, which are known to have large minimal distances as spherical codes. A geometric connection between spherical and…
We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on $N$ qubits with logarithmic weight stabilizers and distance $N^{1-\epsilon}$ for any…
A construction is presented that allows to produce subspace codes of long length using subspace codes of shorter length in combination with a rank metric code. The subspace distance of the resulting code, called linkage code, is as good as…
Construction $C^\star$ was recently introduced as a generalization of the multilevel Construction C (or Forney's code-formula), such that the coded levels may be dependent. Both constructions do not produce a lattice in general, hence the…
Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor $\log\log n$ for infinitely many dimensions $n$. Here we prove an effective version of this result, in the sense that we exhibit, for the same…
Spatially-coupled (SC) codes are a family of graph-based codes that have attracted significant attention thanks to their capacity approaching performance and low decoding latency. An SC code is constructed by partitioning an underlying…
A toric quantum error-correcting code construction procedure is presented in this work. A new class of an infinite family of toric quantum codes is provided by constructing a classical cyclic code on the square lattice $\mathbb{Z}_{q}\times…
Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the…
We obtain algorithmically effective versions of the dense lattice sphere packings constructed from orders in $\mathbb{Q}$-division rings by the first author. The lattices in question are lifts of suitable codes from prime characteristic to…
A heuristic construction of polar codes for successive cancellation list (SCL) decoding with a given list size is proposed to balance the trade-off between performance measured in frame error rate (FER) and decoding complexity. Furthermore,…
Encoding and indexing of lattice codes is generalized from self-similar lattice codes to a broader class of lattices. If coding lattice $\Lambda_{\textrm{c}}$ and shaping lattice $\Lambda_{\textrm{s}}$ satisfy $\Lambda_{\textrm{s}}…
We generalize a construction of non-binary quantum LDPC codes over $\F_{2^m}$ due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve…