In this paper, we investigate the smallest value of p for which a (J,L,p)-QC LDPC code with girth 6 exists for J=3 and J=4. For J=3, we determine the smallest value of p for any L. For J=4, we determine the smallest value of p for L ≤ 301. Furthermore we provide examples of specific constructions meeting these smallest values of p.

In digital content distribution systems, digital watermarking (fingerprinting) technique provides a good solution to avoid illegal copying and has been studied very actively. c-Secure CRT Code is one of the most practical ID coding schemes for such fingerprinting since it is secure against collusion attacks and also secure even though random errors are furthermore added. But its usefulness is decreased in the case that random errors are added because the code length will be longer. In this paper, a new collusion attack with addition of random errors is introduced and show that c-Secure CRT Code is not sufficiently secure against the attack at first. Next, we analyze the problem and propose a new ID coding scheme, Randomized c-Secure CRT Code which overcomes the problem. As a result, this new scheme improves the error tracing probabilities against the proposed attack drastically. This new scheme has the same code length, so this is one of the most responsible fingerprinting codes for content distribution systems.

The code length of Tardos's collusion-secure fingerprint code is of theoretically minimal order with respect to the number of adversarial users (pirates). However, the constant factor should be further reduced for practical implementation. In this article, we improve the tracing algorithm of Tardos's code and propose a 2-secure and short random fingerprint code, which is secure against collusion attacks by two pirates. Our code length is significantly shorter than that of Tardos's code and its tracing error probability is practically small.