Synthetic DNA can in principle be used for the archival storage of arbitrary data. Because errors are introduced during DNA synthesis, storage, and sequencing, an error-correcting code (ECC) is necessary for error-free recovery of the data. Previous work has utilized ECCs that can correct substitution errors, but not insertion or deletion errors (indels), instead relying on sequencing depth and multiple alignment to detect and correct indels -- in effect an inefficient multiple-repetition code. This paper describes an ECC, termed "HEDGES", that corrects simultaneously for substitutions, insertions, and deletions in a single read. Varying code rates allow for correction of up to 10 nucleotide errors and achieve 50 or better of the estimated Shannon limit.
Church et. al @cite_16 synthesized oligomers of length 159, each of which contained both address information (ordering of oligomers in the message) and payload. There was no explicit ECC. The pool of oligomers was sequenced to a depth 3000x, allowing recovery of a consensus sequence with high probability. High-depth coverage can correct sequencing errors, but not synthesis errors. Indeed, the final results contained 10-bit errors.
@cite_16
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The purpose of this paper is to resolve several open problems in the current literature on Boolean circuits, communication complexity, and hashing functions. These lower bound results share the common feature that their proofs utilize probabilistic arguments in an essential way. Specifically, we prove that, to compute the majority function of n Boolean variables, the size of any depth-3 monotone circuit must be greater than 2ne, and the size of any width-2 branching program must have super-polynomial growth. We also show that, for the problem of deciding whether i ≤ j for two n-bit integers i and j, the probabilistic e-error one-way communication complexity is of order θ(n), while the two-way e-error complexity is O((log n)2). We will also prove that, to compute i ? j mod p for an n-bit prime p, the probabilistic e-error two-way communication complexity is of order θ(n). Finally, we prove a conjecture of Ullman that uniform hashing is asymptotically optimal in its expected retrieval cost among open address hashing schemes. Shor and Preskill (see Phys. Rev. Lett., vol.85, p.441, 2000) have provided a simple proof of security of the standard quantum key distribution scheme by Bennett and Brassard (1984) by demonstrating a connection between key distribution and entanglement purification protocols (EPPs) with one-way communications. Here, we provide proofs of security of standard quantum key distribution schemes, Bennett and Brassard and the six-state scheme, against the most general attack, by using the techniques of two-way entanglement purification. We demonstrate clearly the advantage of classical post-processing with two-way classical communications over classical post-processing with only one-way classical communications in quantum key distribution (QKD). This is done by the explicit construction of a new protocol for (the error correction detection and privacy amplification of) Bennett and Brassard that can tolerate a bit error rate of up to 18.9 , which is higher than what any Bennett and Brassard scheme with only one-way classical communications can possibly tolerate. Moreover, we demonstrate the advantage of the six-state scheme over Bennett and Brassard by showing that the six-state scheme can strictly tolerate a higher bit error rate than Bennett and Brassard. In particular, our six-state protocol can tolerate a bit error rate of 26.4 , which is higher than the upper bound of 25 bit error rate for any secure Bennett and Brassard protocol. Consequently, our protocols may allow higher key generation rate and remain secure over longer distances than previous protocols. Our investigation suggests that two-way entanglement purification is a useful tool in the study of advantage distillation, error correction, and privacy amplification protocols. A problem of index coding with side information was first considered by Birk and Kol in 1998. In this study, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols δ , what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the α-bound and the κ -bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced. Several bounds on the length of static ECICs are derived, and constructions for static ECICs are discussed. Connections of these codes to weakly resilient Boolean functions are established. In 2010, Tao Xie and Dengguo Feng [XF10] constructed the rst single-block collision for MD5 consisting of two 64-byte messages that have the same MD5 hash. Details of their attack, developed using what they call an evolutionary approach, has not been disclosed security reasons". Instead they have posted a challenge to the cryptology community to nd a new dierent single-block collision attack for MD5. This paper answers that challenge by presenting a single-block collision attack based on other message dierences together with an example colliding message pair. The attack is based on a new collision nding algorithm that exploits the low number of bitconditions in the rst round. It uses a new way to choose message blocks that satisfy bitconditions up to step 22 and additionally uses three known tunnels to correct bitconditions up to step 25. The attack has an average runtime complexity equivalent to 2 49:8 calls to MD5’s compression function.