Any code C with covering radius R must satisfy a set of linear inequalities that involve the Lloyd polynomial L R (x) ; these generalize the sphere bound. The "syndrome graphs" associated with a linear code C help to keep track of low weight vectors in the same coset of C (if there are too many such vectors C cannot exist). As illustrations it is shown that t[17,10] = 3 and t[23,15] = 3, where ...

For any integer ρ ≥ 1 and for any prime power q, the explicit construction of a infinite family of completely regular (and completely transitive) q-ary codes with d = 3 and with covering radius ρ is given. The intersection array is also computed. Under the same conditions, the explicit construction of an infinite family of q-ary uniformly packed codes (in the wide sense) with covering radius ρ,...

In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound. Based on the asymptotic bounds, we observe that asymptotically Gilbert-Varsharmov bound is exceeded by MRD co...

On the development of flexible electronics, a highly flexible nonvolatile memory, which is an important circuit component for the portability, is necessary. However, the flexibility of existing nonvolatile memory has been limited, e.g. the smallest radius into which can be bent has been millimeters range, due to the difficulty in maintaining memory properties while bending. Here we propose the ...

A binary code with covering radius R is a subset C of the hypercube Qn = {0, 1}n such that every x ∈ Qn is within Hamming distance R of some codeword c ∈ C, where R is as small as possible. For a fixed coordinate i ∈ [n], define C b , for b ∈ {0, 1}, to be the set of codewords with a b in the ith position. Then C is normal if there exists an i ∈ [n] such that for any v ∈ Qn, the sum of the Hamm...

In this paper, we study embedding efficiency, which is an important attribute of steganographic schemes directly influencing their security. It is defined as the expected number of embedded random message bits per one embedding change. Constraining ourselves to embedding realized using linear covering codes (so called matrix embedding), we show that the quantity that determines embedding effici...

The problems of intersection and union of spheres of the same radius in Hamming metric are considered. The formula for number of points in intersection is derived in case of two spheres. It is proved that three or more spheres of radius (covering radius of a code ) centered at points belonging to some quasi-perfect code intersect at most at one point. It is also proved that the increase of card...

A conjecture of Woods from 1972 is disproved. A lattice in R is called well-rounded if its shortest nonzero vectors span R, is called unimodular if its covolume is equal to one, and the covering radius of a lattice Λ is the least r such that R = Λ + Br, where Br is the closed Euclidean ball of radius r. Let Nd denote the greatest value of the covering radius over all well-rounded unimodular lat...

In this paper we present a family of q-ary nonlinear quasi-perfect codes with covering radius 2. The have length $$n = q^m$$ and size $$ M q^{n - m 1}$$ where q is prime power, $$q \ge 3$$ , an integer, $$m 2$$ . We prove that there are more than $$q^{q^{cn}}$$ nonequivalent such n, for all sufficiently large n constant $$c \frac{1}{q} \varepsilon