A locally testable code (LTC) is an error-correcting code that has a property-tester. The tester reads $q$ bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter $q$ is called the locality of the tester. LTCs were initially studied as important components of PCPs, and since then the topic has evolved on its own. High rate LTCs could be useful in practice: before attempting to decode a received word, one can save time by first quickly testing if it is close to the code. An outstanding open question has been whether there exist "$c^3$-LTCs", namely LTCs with *c*onstant rate, *c*onstant distance, and *c*onstant locality. In this work we construct such codes based on a new two-dimensional complex which we call a left-right Cayley complex. This is essentially a graph which, in addition to vertices and edges, also has squares. Our codes can be viewed as a two-dimensional version of (the one-dimensional) expander codes, where the codewords are functions on the squares rather than on the edges.
翻译:本地测试代码( LTC) 是一个有属性值的错误校正代码( LTC) 。 测试者读取的是随机选择的 $q 位元, 拒绝的单词的概率与其与代码的距离成正比。 参数 $q 美元 被称为测试者的位置 。 LTC 最初作为五氯苯酚的重要成分进行了研究, 自此以后, 主题就演变了 。 高率 LTC 在实践中可能有用 : 在尝试解码接收的单词之前, 如果它接近代码, 就可以通过快速测试来节省时间。 一个尚未解决的问题是, 是否存在“ $c%3$- LTC ”, 是否存在“ $c*3$- LTC ” 。 即有 *c* onstant 率的 LTC, *c* onstant 距离, 和 *c* onstant 位置 。 在这项工作中, 我们根据新的二维复合体构建了这样的代码, 我们称之为左右Cayley 复杂体。 这基本上是一张图表, 除了脊边缘和边缘外, 方格外的代码是方格。