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This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single deletion are limited to having an information rate at most 1/2 (achieved by the trivial two fold repetition code). Previously, it was (erroneously) reported that more generally no nontrivial linear codes correcting k deletions exist, i.e., that the (k + 1)-fold repetition codes and its rate of 1/(k + 1) are basically optimal for any k. We disprove this and show the existence of binary linear codes of length n and rate just below 1/2 capable of correcting \Omega(n) insertions and deletions. This identifies rate 1/2 as a sharp threshold for recovery from deletions for linear codes and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We prove novel outer bounds and existential inner bounds for the rate vs. (edit) distance trade-off of linear insdel codes. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically good linear code for Hamming errors into an asymptotically good linear code for insdel errors. Last, we show that the 2 -rate limitation does not hold for affine codes by giving an explicit affine code of rate 1 - e epsilon which can efficiently correct a constant fraction of insdel errors. [ABSTRACT FROM AUTHOR]

Let C denote a binary linear code with length n all of whose coordinates are essential; i.e., for each coordinate there is a codeword that is not zero in that position. Then the maximum distance D is strictly bigger than n/2, and the extremum D = (n + 1)/2 is attained exactly by punctured Hadamard codes. In this paper, we classify binary linear codes with D = n/2 + 1. All of these codes can be produced from punctured Hadamard codes in one of essentially three different ways, each having a transparent description. [ABSTRACT FROM AUTHOR]

Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. In this work, we study a very simple construction of binary linear codes that correct a given number of errors K. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the co de and nearly linear in the number of errors K. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in nonadaptive group testing and construct simple explicit measurement schemes with O(K² log² N) tests and O(K³ log² N) recovery time for identifying up to K defectives in a population of size N. [ABSTRACT FROM AUTHOR]

Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. An important class of lattices are those that possess an orthogonal basis, since if such an orthogonal basis is known, then many other fundamental problems on lattices can be solved easily (e.g., the Closest Vector Problem). However, intriguingly, deciding whether a lattice has an orthogonal basis is not known to be either NP-complete or in P. In this paper, we focus on the orthogonality decision problem for a well-known family of lattices, namely Construction-A lattices. These are lattices of the form C + qℤn, where C is an error-correcting q-ary code, and are studied in communication settings. We provide a complete characterization of lattices obtained from binary and ternary codes using Construction-A that have an orthogonal basis. We use this characterization to give an efficient algorithm to solve the orthogonality decision problem. Our algorithm also finds an orthogonal basis if one exists for this family of lattices. [ABSTRACT FROM AUTHOR]

We investigate the maximal asymptotic rates of length-n binary codes with VC-dimension at most dn and minimum distance at least δn. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining the Sauer--Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert--Varshamov-type arguments over constant-weight and Markov-type sets. [ABSTRACT FROM AUTHOR]

The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees, and a huge subclass of random grid trees containing random binary search trees, random median-of-(2k+1) search trees, random m-ary search trees, random quadtrees, random simplex trees, etc. An important class of random grid trees for which the Wiener index was not studied so far is random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in the affirmative two questions posed by Neininger. [ABSTRACT FROM AUTHOR]