Related papers: Lattice (List) Decoding Near Minkowski's Inequalit…

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete structure of linear codes and point lattices in R^N, and their…

We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits…

In the study of Euclidean lattices, the product of the successive minima is bounded from above and below by explicit quantities. This result is known as Minkowski's second theorem, and can be refined to include Hermite's constant in the…

We show a $2^{n/2+o(n)}$-time algorithm that finds a (non-zero) vector in a lattice $\mathcal{L} \subset \mathbb{R}^n$ with norm at most $\tilde{O}(\sqrt{n})\cdot \min\{\lambda_1(\mathcal{L}), \det(\mathcal{L})^{1/n}\}$, where…

We give a deterministic O(log n)^n algorithm for the {\em Shortest Vector Problem (SVP)} of a lattice under {\em any} norm, improving on the previous best deterministic bound of n^O(n) for general norms and nearly matching the bound of…

Given a lattice $\Lambda \subset \mathbb{R}^n$, we consider its Minkowski reduced basis and the solid angle $\Omega$ spanned by the basis vectors. Such a basis satisfies strong near-orthogonality conditions, which allow us to bound from…

Finding the shortest vectors in a lattice is an NP-hard problem, so low-dimensional results also play an essential role in lattice reduction theory. Using Ryskov's result for the admissible centerings and Tammela's result for determining…

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…

We compare for an $n$-dimensional Euclidean lattice $\Lb$ the smallest possible values of the product of the norms of $n$~vectors which either constitute a basis for $\Lb$ (Hermite-type inequalities) or are merely assumed to be independent…

Reed--Solomon error-correcting codes are ubiquitous across computer science and information theory, with applications in cryptography, computational complexity, communication and storage systems, and more. Most works on efficient error…

LDPC lattices were the first family of lattices that equipped with iterative decoding algorithms under which they perform very well in high dimensions. In this paper, we introduce quasi cyclic low density parity check (QC-LDPC) lattices as…

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…

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice…

We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming…

Since the invention of the famous LLL algorithm, lattice reduction has been an extremely useful tool in computational number theory. By construction, the LLL algorithm deals with lattices living in a vector space endowed with a positive…

The Erd\H{o}s distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$…

Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimum positive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ... + e_k \sqrt{s_k} -t | $$ where $ s_1, s_2, ..., s_k$ are positive integers no larger than $n$, $t$ is an…

We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e.,…

The lattice $A_n^*$ is an important lattice because of its covering properties in low dimensions. Clarkson \cite{Clarkson1999:Anstar} described an algorithm to compute the nearest lattice point in $A_n^*$ that requires $O(n\log{n})$…

We show that the number of lattice directions in which a d-dimensional convex body in R^d has minimum width is at most 3^d-1, with equality only for the regular cross-polytope. This is deduced from a sharpened version of the 3^d-theorem due…