Related papers: A ternary construction of lattices
We present a new construction of triple arrays by combining a symmetric 2-design with a resolution of another 2-design. This is the first general method capable of producing non-extremal triple arrays. We call the triple arrays which can be…
We propose a coding scheme that achieves the capacity of the compound MIMO channel with algebraic lattices. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned…
In a recent paper [M. Araya, M. Harada, Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes, Des. Codes Cryptogr., 91 (2023), 1813--1843], Araya and Harada gave…
After a short review of the Method of Recursive Counting we introduce a general algebraic description of recursive lattice building. This provides a rigorous framework for discussion of method's limitations.
An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low…
If $L$ is a finite lattice, we show that there is a natural topological lattice structure on the geometric realization of its order complex $\Delta(L)$ (definition recalled). Lattice-theoretically, the resulting object is a subdirect…
Shimada and Zhang studied the existence of polarizations on some supersingular $K3$ surfaces by reducing the existence of the polarizations to that of ternary $[12,5]$ codes satisfying certain conditions. In this note, we give a…
We give new results on the structure and representations of the three lattices $\mathbf{\Lambda }_{\mathrm{k}},\mathbf{\Lambda }_{\mathrm{k}\mathcal{C}},\mathbf{\Lambda }_{\mathrm{k}}^{\ast }$ relevant to code CFTs realizing Narain…
Given a fixed integer $n\geq 2$, we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattice and a quotient lattice of the cyclic Dyer lattice, which…
The Narain lattice construction of string compactifications is generalized to include spontaneously broken supersymmetry. Consistency conditions from modular invariance and Lorentz symmetry are solved in full generality. This framework…
We propose a lattice construction from totally real and CM fields, which naturally generalizes the Construction A of lattices from $p$-ary codes obtained from the cyclotomic field $\mathbb{Q}(\zeta_p)$, $p$ a prime, which in turn contains…
The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite…
We present an easy construction producing a Kleene lattice K from an arbitrary distributive lattice L and a non-empty subset of L. We show that L can be embedded into K and compute the cardinality of K under certain additional assumptions.…
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and…
A weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight $p^m$ is constructed and shown that the rows of $W$ and $-W$ form optimal constant weight ternary codes of length $n$, weight $p^m$ and minimum distance…
We construct an infinite series of simply transitive irreducible lattices in PGL_2(F_q((t))) \times PGL_2(F_q((t))) by means of a quaternion algebra over F_q(t). The lattices depend on an odd prime power q = p^r and a parameter \tau\ in…
We construct new linear codes with high minimum distance d. In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n,k. Among these new codes there is an optimal ternary…
In this work, we give three new techniques for constructing Hermitian self-dual codes over commutative Frobenius rings with a non-trivial involutory automorphism using $\lambda$-circulant matrices. The new constructions are derived as…
Three classes of binary linear codes with at most four nonzero weights were constructed in this paper, in which two of them are projective three-weight codes. As applications, $s$-sum sets for any odd $ s > 1$ were constructed.
In this note we consider ternary trees naturally embedded in the plane in a deterministic way such that the root has position zero, or in other words label zero, and the children of a node with position $j$ have positions $j-1$, $j$, and…