Perturbation Codes: A New Class of Linear Convolutional Codes

扰动码：一类新的线性卷积码

• 批准号: 0830381
• 负责人: John Kieffer
• 金额: $ 11.5万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830381&HistoricalAwards=false
• 关键词: Perturbation; Codes; New; Class; Linear

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.

NSF 提案 0830381 的摘要项目名称：扰动码：一类新型线性卷积码线性卷积码广泛应用于数据通信系统和数据存储系统中。每个这样的代码都以固定的速率和固定的延迟处理数据，因此重构数据中的错误比例（称为平均失真）将是可以接受的小值。线性卷积码设计的首要问题是找到一个最优码，即在有限个可行码集合中产生最小平均失真的码。在大多数情况下，使用过去的技术，这个问题只能近似解决，产生几乎最优的代码（具有接近最小平均失真的代码），可行的代码集合太大，无法单独检查代码的平均失真。在这个项目中，我们研究了一种称为扰动理论的新技术，它将可行代码的集合减少到一个更小的代码集合，称为扰动代码。在许多情况下，最优代码将是扰动代码之一，并且可以在合理的时间内找到它。可行代码集中的每个线性卷积代码都通过二进制字段上的奇偶校验矩阵来描述，该矩阵由固定数量的行和列组成。扰动类代码由其奇偶校验矩阵可以排列成使得顶部的一定行数保持固定的代码组成。任何代码都将位于多个扰动类代码中，具体取决于奇偶校验矩阵的哪些行保持固定。在某些条件下，将存在一个自然群作用于扰动类代码的奇偶校验矩阵。使用组结构，选择每个扰动类代码的子集。如果某个代码属于包含该代码的一定数量的扰动类中的选定代码，则该代码将被声明为扰动代码。扰动代码的定义具有灵活性。在这个项目中，我们研究如何正确描述扰动码概念，以便可以从一小组扰动码中找到最佳代码，以获得足够对称的源和通道模型。