Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $n$ and a prime power $q$. Based on Pascal's triangle we give an explicit construction of universally decodable matrices for any non-zero integers $L$ and $n$ and any prime power $q$ where $L \leq q+1$. This is the largest set of possible parameter values since for any list of universally decodable matrices the value $L$ is upper bounded by $q+1$, except for the trivial case $n = 1$. For the proof of our construction we use properties of Hasse derivatives, and it turns out that our construction has connections to Reed-Solomon codes, Reed-Muller codes, and so-called repeated-root cyclic codes. Additionally, we show how universally decodable matrices can be modified so that they remain universally decodable matrices.
Coded computation is an emerging research area that leverages concepts from erasure coding to mitigate the effect of stragglers (slow nodes) in distributed computation clusters, especially for matrix computation problems. In this work, we present a class of distributed matrix-vector multiplication schemes that are based on codes in the Rosenbloom-Tsfasman metric and universally decodable matrices. Our schemes take into account the inherent computation order within a worker node. In particular, they allow us to effectively leverage partial computations performed by stragglers (a feature that ma…
In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that may consist of duplicate entries. We first introduce a new class of matrices called multipermutation matrices. We characterize the convex hull of multipermutation matrices. Based on this characterization, we propose a new class of codes that we term LP-decodable multipermutation codes. Then, we derive two LP decoding algorithms. We first formulate an LP decoding problem for memoryless channels. We then derive an LP algorithm that minimizes the Chebysh…
In this paper, we introduce a new way of constructing and decoding multipermutation codes. Multipermutations are permutations of a multiset that generally consist of duplicate entries. We first introduce a class of binary matrices called multipermutation matrices, each of which corresponds to a unique and distinct multipermutation. By enforcing a set of linear constraints on these matrices, we define a new class of codes that we term LP-decodable multipermutation codes. In order to decode these codes using a linear program (LP), thereby enabling soft decoding, we characterize the convex hull o…
The design of linear STBCs offering a low-complexity ML decoding using the well known Sphere Decoder (SD) has been extensively studied in last years. The first considered approach to derive design criteria for the construction of such codes is based on the Hurwitz-Radon (HR) Theory for mutual orthogonality between the weight matrices defining the linear code. This appproach served to construct new families of codes admitting fast sphere decoding such as multi-group decodable, fast decodable, and fast-group decodable codes. In a second Quadratic Form approach, the Fast Sphere Decoding (FSD) com…
Fast-Decodable Asymmetric Space-Time Codes from Division Algebras
多输入双输出（MIDO）码在不久的将来的无线通信中是重要的，其中便携式终端用户设备物理上很小并且通常将包含至多两个接收天线。特别诱人的是4 x 2通道，由于其在数字视频广播（DVB）中的直接适用性。这种信道最佳地采用由（4 × 4）矩阵组成的速率2空时（ST）码。不幸的是，这样的代码通常解码起来非常复杂，因此提出了对具有降低的复杂性的构造的要求。最近，已经提出了一些降低复杂性的结构，但它们主要是基于不同的特设方法，并导致孤立的例子，而不是在一个更一般的代码类。在本文中，它将被证明，一个家庭的划分代数为基础的MIDO码将始终导致至少37.5%的最坏情况下的复杂性降低，同时保持充分的多样性，并首次，非零行列式（NVD）属性。减少遵循的事实是，类似于阿拉姆码，代码将是哈密尔顿四元数的矩阵环的子集，因此允许简化解码。目前，这种减少是最有名的速率二MIDO码。通过计算机模拟，给出了几种显式结构，并证明了它们具有优良的性能。
