Classification of Optimal Linear Codes

9801581 Jaffe这个奖项支持对一个项目的研究，该项目涉及搜索和实现最佳线性码分类的算法，重点是二进制情况。这里包括发现新的最优线性码并以有意义的方式描述它们的问题。所有这些都是正在进行的计算机语言Split开发的一部分，该语言Split促进了关于线性代码的证明的机械验证。这项研究涉及最优纠错码，它在现代技术中扮演着无处不在的角色，因为它们使得在噪声信道上传输信息成为可能。例如，这种代码促进了地球-卫星通信(尽管受到大气干扰)，这种代码促进了光盘的读取(尽管光盘表面有灰尘和损坏)。代码是抽象的数学对象。那些对传递信息最有效的被称为最佳。这是一个悬而未决的数学问题--也是这项研究的主题--来确定代码的有效性。这个问题可以转化为一个被称为线性规划的计算问题，许多行业(如航空业)也使用线性规划来优化其资源使用。由于现有的线性规划软件不能很好地回答编码理论中出现的问题，将开发新的软件。此外，允许对代码断言进行机械验证的软件将为这些线性规划计算提供“前端”。所有这类软件(包括源代码)都将免费提供给公众。

9801581 Jaffe This award supports research on a project involving the search for and implementation of alogorithms for classifying optimal linear codes, with an emphasis on the binary case. Included here is the problem of discovering new optimal linear codes and describing them in a meaningful way. All of this is part of the ongoing development of a computer language Split, which facilitates the mechanical verification of proofs about linear codes. This research concerns optimal error-correcting codes, which play a ubiquitous role in modern technology, because they make possible the transmission of information across noisy channels. For example, such codes facilitate earth-satellite communications (in spite of atmospheric disturbances), and such codes facilitate the reading of compact disks (in spite of dust and damage to the disk's surface). Codes are abstract mathematical objects. Those that are are most effective for transmitting information are called optimal. It is an open mathematical problem - and the subject of this research - to determine just how effective a code can be. This problem can be converted into a computational problem called "linear programming", which is also used by many industries (such as the airline industry) to optimize their use of resources. As existing linear programming software is not good enough to answer the problems arising from coding theory, new software will be developed. In addition, software which allows for the mechanical verification of assertions about codes will provide a "front end" to these linear programming calculations. All such software (including source code) will be made freely available to the public.