Mathematical Sciences: Binary Linear Codes: Nonexistence and Classification Theorems

9623205 Jaffe该奖项为一个涉及搜索和实现二进制线性码分类算法的项目提供资金。 这些算法被编码在一种语言中，正在开发中，这有利于二进制线性代码的证明的机械验证。一种方法是取一个广义权枚举数，并使用它来生成一个线性规划问题，就像Delsarte对普通权所做的那样 枚举器。 这些线性规划问题往往很大，因此一般不适合手工计算。 一个特别有前途的权重枚举器是分裂权重枚举器，它是由n的划分产生的，其中n是代码的长度。 利用这种方法和其他方法，将证明关于二元线性码的许多新的具体结果。 反过来，这些可能会建议新的算法和封闭形式的一般陈述。 本研究是关于错误校正码的，它是数学组合学、计算机科学和电子工程中的一个基本课题。 当信息必须通过嘈杂的信道从一个地方传送到另一个地方时，这种代码是必不可少的。 例如，它们被用在光盘播放器中，将数字化的音乐从光盘的表面传送到播放器的内部，以及用于地球和地球之间的通信。 轨道卫星 纠错码的工作原理是创造性地在传输的信息中加入冗余，使接收者能够检测和纠正传输过程中出现的错误，这些错误是由CD表面的灰尘或宇宙射线等引起的。如果一个代码允许检测和纠正许多错误，而不产生过多冗余的代价，那么它就是好的。 该奖项既资助了对好代码的搜索，也资助了对它们能有多好的数学限制的调查。

9623205 Jaffe This award provides funding for a project involving the search for and implementation of algorithms for classifying binary linear codes. These algorithms are codified in a language, under development, which facilitates the mechanical verification of proofs about binary linear codes. One method is to take a generalized weight enumerator, and use it to generate a linear programming problem, as Delsarte did with the ordinary weight enumerator. Thse linear programming problems tend to be large, and thus are not generally amenable to hand calculation. One particularly promising weight enumerator is the split weight enumerator arising from a partition of n, where n is the length of the code. Using this method and others, many new specific results about binary linear codes will be proved. In turn, these may suggest new algorithms and closed-form general statements. This research is about error-correcting codes, a fundamental topic in mathematical combinatorics, computer science, and electrical engineering. Such codes are indispensable whenever information must be transferred across noisy channels from one place to another. For example,they are used in compact disk players to get digitized music from the surface of the CD to the inside of the player, and in communications between the earth and orbiting satellites. Error-correcting codes work by creatively incorporating redundancy into the transmitted message, to permit the receiver to detect and correct those errors introduced in transit, which were caused by (for instance) dust on the surface of the CD or cosmic rays. A code is good if it allows the detection and correction of many errors without incurring the expense of excessive redundancy. This award funds both a search for good codes and an investigation into the mathematical limitations of how good they can be.