Perturbation Codes: A New Class of Linear Convolutional Codes

美国国家科学基金会提案摘要0830381项目标题：微扰码：一类新的线性卷积码线性卷积码在数据通信系统和数据存储系统中广泛存在。每个这样的代码以固定的速率和固定的延迟处理数据，因此重建数据中的误差分数(称为平均失真)将是可以接受的小的。线性卷积码设计的首要问题是找到一个最优码，它是在一组可行的码中产生最小平均失真的码。在大多数情况下，用过去的技术只能近似地解决这个问题，产生一个几乎最优的码(具有接近最小平均失真的码)，可接受码的集合太大而不能单独检查码的平均失真。在这个项目中，我们研究了一种新的技术，称为扰动理论，它将可行码的集合减少为称为扰动码的更小的码集。在许多情况下，最优码将是扰动码之一，并且可以在合理的时间量内找到它。该组可行码中的每个线性卷积码通过二进制域上的奇偶校验矩阵来描述，该二进制域由固定数量的行和列组成。扰动码类由其奇偶校验矩阵可以被排列的码组成，使得在顶部的一定数量的行保持X。根据奇偶校验矩阵的哪些行被保持，任何码都将位于一个以上的码扰动类中。在某些条件下，将存在作用于扰动码类的奇偶校验矩阵的自然群。使用组结构，选择每个扰码类的子集。如果一个码是包含该码的一定数量的扰动类中的所选码之一，则该码将被宣布为扰动码。摄动码的定义存在一定的容错性。在这个项目中，我们研究了如何正确地描述扰动码的概念，以便从一小部分扰动码中找到一个最优码，用于表面对称的信源和信道模型。

Abstract for NSF Proposal 0830381Project Title: Perturbation Codes: A New Class of LinearConvolutional CodesLinear convolutional codes are widespread in data communication systems and datastorage systems. Each such code processes data at a ¯xed rate and with a ¯xed delay, so thatthe fraction of errors (called average distortion) in the reconstructed data will be acceptablysmall. The primary problem of linear convolutional code design is to ¯nd an optimal code,which is a code yielding the minimum average distortion among codes in a ¯nite set of feasiblecodes. In most cases, with past techniques, this problem could only be solved approximately,yielding an almost optimal code (a code with near minimum average distortion), the set offeasible codes being too large for the average distortion of codes to be examined individually.In this project, we investigate a new technique, called perturbation theory, which reduces theset of feasible codes to a much smaller set of codes called perturbation codes. In many cases,an optimal code will be one of the perturbation codes, and it can be found in a reasonableamount of time.Each linear convolutional code in the set of feasible codes is described via a parity checkmatrix over the binary ¯eld, consisting of a ¯xed number of rows and columns. A perturba-tion class of codes consists of codes whose parity check matrices can be arranged so that acertain number of rows at the top remain ¯xed. Any code will lie in more than one pertur-bation class of codes, depending upon which rows of the parity check matrix are held ¯xed.Under certain conditions, there will be a natural group acting on the parity check matricesof a perturbation class of codes. Using the group structure, a subset of each perturbationclass of codes is selected. A code will be declared to be a perturbation code if it is among theselected codes in a certain number of perturbation classes containing the code. There is °ex-ibility in the de¯nition of perturbation code. In this project, we investigate how to properlydelineate the perturbation code concept so that an optimal code can be found from amonga small set of perturbation codes, for source and channel models of su±cient symmetry.