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Robust Parent-Identifying Codes and Combinatorial Arrays

机译：健壮的父母识别码和组合数组

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摘要

An $n$-word $y=(y_{1},dots, y_{n})$ over a finite alphabet of cardinality $q$ is called a descendant of a set of $t$ words $x^{1},dots, x^{t}$ if every coordinate $y_{i},i=1,dots, n$, is contained in the set ${x^{1}_{i},dots, x^{t}_{i}}$. A code $ {cal C}={x^{1},dots, x^{M}}$ is said to have the $t$-IPP property if for any $n$ -word $y$ that is a descendant of at most $t$ parents belonging to the code, it is possible to identify at least one of them. From earlier works, it is known that $t$-IPP codes of positive rate exist if and only if $tleq q-1$. We introduce a robust version of IPP codes which allows error-free identification of parents in the presence of a certain number of mutations, i.e., coordinates in $y$ that can break away from the descent rule, taking arbitrary values from the alphabet or becoming completely unreadable. We show existence of robust $t$-IPP codes for all - tex Notation="TeX">$tleq q-1$ and some positive proportion of such coordinates. We uncover a relation between the hash distance of codes and the IPP property and use it to find the exact proportion of mutant coordinates that permits identification of pirates with zero probability of error in the case of size-2 coalitions.
机译：在基数为$ q $的有限字母上的$ n $单词$ y =（y_ {1}，dots，y_ {n}）$被称为一组$ t $单词$ x ^ {1}的后代，dots，x ^ {t} $如果每个坐标$ y_ {i}，i = 1，dots，n $，包含在集合$ {x ^ {1} _ {i}，dots，x ^ {t中} _ {i}} $。如果对于任何作为后代的$ n $ -word $ y $，则代码$ {cal C} = {x ^ {1}，dots，x ^ {M}} $被称为具有$ t $ -IPP属性。在属于该代码的最多$ t $个父母中，有可能识别出其中至少一个。从更早的著作中知道，当且仅当$ tleq q-1 $时，存在$ t $ -IPP的正利率代码。我们引入了一个强大的IPP代码版本，该代码可以在存在一定数量的突变的情况下对父母进行无误识别，即$ y $坐标可以脱离后裔规则，从字母中取任意值或变为完全不可读。我们显示了所有健壮的$ t $ -IPP代码的存在-tex Notation =“ TeX”> $ tleq q-1 $和此类坐标的一些正比例。我们发现了代码的哈希距离与IPP属性之间的关系，并使用它来找到突变坐标的确切比例，该比例允许在大小为2的联盟的情况下以零错误概率识别海盗。

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