Classification of Optimal Linear Codes

小行星9801581 该奖项支持一个项目的研究，该项目涉及搜索和实现用于分类最佳线性代码的算法，重点是二进制情况。这里包括的问题是发现新的最优线性码，并描述他们在一个有意义的方式。所有这些都是正在开发的计算机语言Split的一部分，该语言有助于对线性代码的证明进行机械验证。 本研究涉及最优纠错码，它在现代技术中发挥着无处不在的作用，因为它们使信息在噪声信道中的传输成为可能。 例如，这样的代码便于地球-卫星通信（尽管有大气干扰），并且这样的代码便于光盘的阅读（尽管光盘表面有灰尘和损坏）。代码是抽象的数学对象。 那些对信息传输最有效的被称为最优的。 这是一个开放的数学问题，也是本研究的主题，以确定代码的有效性。 这个问题可以转换成一个计算问题，称为“线性规划”，这也是由许多行业（如航空业），以优化其资源的使用。 由于现有的线性规划软件不足以解决编码理论所带来的问题，因此将开发新的软件。 此外，允许对关于代码的断言进行机械验证的软件将为这些线性编程计算提供“前端”。 所有这些软件（包括源代码）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.