Comment on "New Results on Frame-Proof Codes and Traceability Schemes"

In the paper "New Results on Frame-Proof Codes and Traceability Schemes" by Reihaneh Safavi-Naini and Yejing Wang [IEEE Trans. Inform. Theory, vol. 47, no. 7, pp. 3029-3033, Nov. 2001], there are lower bounds for the maximal number of codewords in binary frame-proof codes and decoders in traceability schemes. There are also existence proofs using a construction of binary frame-proof codes and traceability schemes. Here it is found that the main results in the referenced paper do not hold.Comment: 3 page

Improved Constructions of Frameproof Codes

Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let $M_{c,l}(q)$ be the largest cardinality of a $q$-ary $c$-frameproof code of length $l$ and $R_{c,l}=\lim_{q\rightarrow \infty}M_{c,l}(q)/q^{\lceil l/c\rceil}$. It has been determined by Blackburn that $R_{c,l}=1$ when $l\equiv 1\ (\bmod\ c)$, $R_{c,l}=2$ when $c=2$ and $l$ is even, and $R_{3,5}=5/3$. In this paper, we give a recursive construction for $c$-frameproof codes of length $l$ with respect to the alphabet size $q$. As applications of this construction, we establish the existence results for $q$-ary $c$-frameproof codes of length $c+2$ and size $\frac{c+2}{c}(q-1)^2+1$ for all odd $q$ when $c=2$ and for all $q\equiv 4\pmod{6}$ when $c=3$. Furthermore, we show that $R_{c,c+2}=(c+2)/c$ meeting the upper bound given by Blackburn, for all integers $c$ such that $c+1$ is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o

放送型暗号の組合せ的構造及びマルチメディア指紋符号に関する進展

筑波大学 (University of Tsukuba)201

Construction of Almost Disjunct Matrices for Group Testing

In a \emph{group testing} scheme, a set of tests is designed to identify a small number $t$ of defective items among a large set (of size $N$) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a \emph{disjunct matrix}, an $M \times N$ matrix where the union of supports of any $t$ columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number $M$ of rows (tests). One of the main ways of constructing disjunct matrices relies on \emph{constant weight error-correcting codes} and their \emph{minimum distance}. In this paper, we consider a relaxed definition of a disjunct matrix known as \emph{almost disjunct matrix}. This concept is also studied under the name of \emph{weakly separated design} in the literature. The relaxed definition allows one to come up with group testing schemes where a close-to-one fraction of all possible sets of defective items are identifiable. Our main contribution is twofold. First, we go beyond the minimum distance analysis and connect the \emph{average distance} of a constant weight code to the parameters of an almost disjunct matrix constructed from it. Our second contribution is to explicitly construct almost disjunct matrices based on our average distance analysis, that have much smaller number of rows than any previous explicit construction of disjunct matrices. The parameters of our construction can be varied to cover a large range of relations for $t$ and $N$.Comment: 15 Page

Characterizing the strongly jump-traceable sets via randomness

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $\Delta^0_2$ 1-random sets.Comment: 41 page