集合成形理论在局部可测码开发中的应用
Use of Set Shaping theory in the development of locally testable codes
摘要：
在开发本地可测试代码时，通过在码字中创建冗余来将信息添加到编码消息。在这篇文章中，我们提出了一种替代方法，在该方法中，冗余被引入到必须在编码之前传输的消息中，而不是在码字中。这种方法利用了集合成形理论，其目标是研究双射函数f（X）=Y，该函数将一组字符串转换为由更长的字符串组成的相等大小的集合。通过这种方式，这种类型的函数将自变量x转换为因变量y，其发射概率由先前发射的变量决定。因此，如果解码器解码与等于零的条件概率相关联的符号，则我们检测到消息中的错误。如果使用的函数f是使平均信息含量最小化的函数，则我们开发出可以有效测试的代码。事实上，就压缩而言，可以观察到，更长的字符串被选择具有更小熵的字符串的事实所补偿。
英文摘要：
In developing locally testable codes, information is added to the coded message by creating redundancy in the codewords. In this article, we propose an alternative method in which redundancy is introduced on the message that must be transmitted before its encoding and not in the codewords. This approach exploits the Set Shaping Theory whose goal is the study the bijection functions f(X)=Y that transform a set of strings into a set of equal size made up of strings of greater length. In this way, this type of function transforms the independent variable x into the dependent variable y whose emission probability is conditioned by the previously emitted variables. Thus, if the decoder decodes a symbol associated with a conditional probability equal to zero, we detect an error in the message. If the function f used is the one that minimizes the average information content, we develop a code that can be tested efficiently. In fact, it is observed, in terms of compression, that the greater length of the strings is compensated by the fact of having chosen the strings with less entropy.
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