A λ-fold r-packing (multiple radius- r covering) in a Hamming metric space is a code C such that the radius- r balls centered in the codewords of C cover each vertex of the space by not more (not less, respectively) than λ times. The well-known r-error-correcting codes correspond to the case λ = 1, while in general multifold r-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a q-ary 2-fold 1-packing as q grows; (b) prove that a q-ary distance-2 MDS code of length n is an optimal n-fold 1-packing if q ≥ 2n; (c) derive an upper bound for the size of a binary λ-fold 1-packing and a lower bound for the size of a binary multiple radius-1 covering (the last bound allows to update the small-parameters table); (d) classify all optimal binary 2-fold 1-packings up to length 9, in particular, establish the maximum size 96 of a binary 2-fold 1-packing of length 9; (e) prove some properties of 1-perfect unitrades, which are a special case of 2-fold 1-packings.

A λ-fold r-packing in a Hamming metric space is a code C such that the radius-r balls centered in C cover each vertex of the space by not more than λ-times. The well-known r- error-correcting codes correspond to the case λ = 1. We propose asymptotic bounds for q-ary 2-fold 1-packings as q grows, find that the maximum size of a binary 2-fold 1-packing of length 9 is 96, and derive upper bounds for the size of a binary λ-fold 1 -packing.

In this paper, we present a novel data-hiding method that does not interfere with other data-hiding techniques (e.g., sign bits hiding) that are already included into state-of-the-art coding standards such as HEVC/H.265. One of the main features that are inherent to the proposed technique is its orientation on hierarchically-structured units (e.g., a hierarchy in HEVC/H.265 that includes coding, prediction and transform units). As shown in the paper, this method provides higher coding gain when applied to scalar-quantized values. Finally, we present experimental results that confirm the high RD-performance of this technique in comparison with explicit signaling and discuss its suitability for HEVC-compatible watermarking.

Perfect codes obtained by the Vasil'ev-Schönheim construction from a linear base code and quadratic switching functions are transitive and, moreover, propelinear. This gives at least exp(cN2) propelinear 1-perfect codes of length N over an arbitrary finite field, while an upper bound on the number of transitive codes is exp(C(NlnN)2\vphantom)).

We characterize the set of all n-ary quasigroups of order 4: every n-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an n-ary quasigroup can be represented as a composition of k-ary and $(n-k+1)$-ary quasigroups for some k from 2 to $n-1$, where the order of arguments in the representation can differ from the original order. The set of semilinear n-ary quasigroups has a characterization in terms of Boolean functions.

We can encode rare events with an overhead of about 1.56 bits/event. The contribution of the overhead to the total length of the code is negligible. The encoding and decoding time counted in operations over bits per bit of the code does not depend on the number of appearances of the events. We also present algorithms of the same speed which enumerate the pyramid cubic lattices with an overhead of about 1.56 bits/dimension. The overhead is the price for reaching the ultimate (to within a constant factor) encoding and decoding speed.