Upper bounds are derived for codes in Stiefel and Grassmann manifolds with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces. The new bounds are asymptotically better than the previously known ones.

A new lower bound on the distance distribution of spherical codes is derived. This yields two new upper bounds on the reliability function of the Gaussian channel. These bounds outperform previously known bounds, and imply a new range of rates for which the exact value of the reliability function is known.

Upper bounds are derived for codes in the Grassmann manifold with given minimum chordal distance. They stem from upper bounds for codes in the product of unit spheres and projective spaces. The new bounds are asymptotically better than the previously known ones.

New bounds on the rate of non-binary codes and non-binary constant weight codes are derived. The asymptotic forms of these bounds outperform known bounds in a wide range of distances. The method is based on analysis of subsets in products of Hamming and Johnson association schemes

A new lower bound on the distance distribution of spherical codes is derived. This yields two new upper bounds on the reliability function of the Gaussian channel. These bounds outperform previously known bounds, and imply a new range of rates for which the exact value of the reliability function is known

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. These bounds are based on combinatorial arguments and linear programming. They improve on the previous bounds due to Burshtein et al. It is proved that at least for high rate LDPC codes have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound

For $z_1,z_2,z_3\in\Z^2$, the tristance $d_3(z_1,z_2,z_3)$ is a generalization of the $L_1$-distance on $\mathbb{Z}^2$ to a quality that reflects the relative dispersion of three points rather than two. In this paper we prove that at least 3k 2 colors are required to color the points of $\mathbb{Z}^2$, such that the tristance between any three distinct points, colored with the same color, is at least 4k. We prove that 3k 2 +3k+1 colors are required if the tristance is at least 4k+2. For the first case we show an infinite family of colorings with colors and conjecture that these are the only colorings with 3k 2 colors.

An identifying code C is a subset of the vertices of the square grid ${\mathbb Z}^2$ with the property that for each element v of ${\mathbb Z}^2$, the collection of elements from C at a distance of at most one from v is nonempty and distinct from the collection of any other vertex. We prove that the minimum density of C within ${\mathbb Z}^2$ is $\frac{7}{20}$.