The first examples of perfect $e$-error correcting $q$-ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$-ary codes. The case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1-error correcting codes is still an open problem.
   This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1-error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.

The linear programming method is developed in the space of unitary matrices in order to obtain bounds for unitary codes relative to the so-called diversity sum and diversity product. Theoretical and numerical results improving previously known bounds are derived.

In the article we shall try to give an overview of the interplay between the design of public key cryptosystems and algorithmic arithmetic geometry. We begin in Section 2 with a very abstract setting and try to avoid all structures which are not necessary for protocols like Diffie-Hellman key exchange, ElGamal signature and pairing based cryptography (e.g. short signatures). As an unavoidable consequence of the generality the result is difficult to read and clumsy. But nevertheless it may be worthwhile because there are suggestions for systems which do not use the full strength of group structures (see Subsection 2.2.1) and it may motivate to look for alternatives to known group-based systems.
  But, of course, the main part of the article deals with the usual realization by discrete logarithms in groups, and the main source for cryptographically useful groups are divisor class groups.
  We describe advances concerning arithmetic in such groups attached to curves over finite fields including addition and point counting which have an immediate application to the construction of cryptosystems.
  For the security of these systems one has to make sure that the computation of the discrete logarithm is hard. We shall see how methods from arithmetic geometry narrow the range of candidates usable for cryptography considerably and leave only carefully chosen curves of genus $1$ and $2$ without flaw.
  A last section gives a short report on background and realization of bilinear structures on divisor class groups induced by duality theory of class field theory, the key concept here is the Lichtenbaum-Tate pairing.

We present characterization results on weighted minihypers. We prove the weighted version of the original results of Hamada, Helleseth, and Maekawa. Following from the equivalence between minihypers and linear codes meeting the Griesmer bound, these characterization results are equivalent to characterization results on linear codes meeting the Griesmer bound.

We propose a public-key encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$_p,n$(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$. This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

Let $\mathcal A$_R,q denote a family of covering codes, in which the covering radius $R$ and the size $q$ of the underlying Galois field are fixed, while the code length tends to infinity. The construction of families with small asymptotic covering densities is a classical problem in the area of Covering Codes.
   In this paper, infinite sets of families $\mathcal A$_R,q, where $R$ is fixed but $q$ ranges over an infinite set of prime powers are considered, and the dependence on $q$ of the asymptotic covering densities of $\mathcal A$_R,q is investigated. It turns out that for the upper limit $\mu$_q^*(R,$\mathcal A$_R,q) of the covering density of $\mathcal A$_R,q, the best possibility is $\mu$_q^*(R,$\mathcal A$_R,q)=$O(q)$. The main achievement of the present paper is the construction of optimal infinite sets of families $\mathcal A$_R,q, that is, sets of families such that relation $\mu$_q^*(R,$\mathcal A$_R,q)=$O(q)$ holds, for any covering radius $R\geq 2$.
   We first showed that for a given $R$, to obtain optimal infinite sets of families it is enough to construct $R$ infinite families $\mathcal A$_R,q⁽⁰⁾, $\mathcal A$_R,q⁽¹⁾, $\ldots$, $\mathcal A$_R,q^(R-1) such that, for all $u\geq u$₀, the family $\mathcal A$_R,q^($\gamma$) contains codes of codimension $r$_u$=Ru + \gamma$ and length $f$_q^($\gamma$)($r$_u) where $f$_q^($\gamma$)$(r)=O(q$^(r-R)/R$)$ and $u$₀ is a constant. Then, we were able to construct the necessary families $\mathcal A$_R,q^($\gamma$) for any covering radius $R\geq 2$, with $q$ ranging over the (infinite) set of $R$-th powers. A result of independent interest is that in each of these families $\mathcal A$_R,q^($\gamma$), the lower limit of the covering density is bounded from above by a constant independent of $q$.
   The key tool in our investigation is the design of new small saturating sets in projective spaces over finite fields, which are used as the starting point for the $q$^m-concatenating constructions of covering codes. A new concept of $N$-fold strong blocking set is introduced. As a result of our investigation, many new asymptotic and finite upper bounds on the length function of covering codes and on the smallest sizes of saturating sets, are also obtained. Updated tables for these upper bounds are provided. An analysis and a survey of the known results are presented.

The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve. Hence, the CFVZ cryptosystem has no advantages over traditional elliptic curve cryptography and should not be used in practice.

We characterize all $q$-ary linear completely regular codes with covering radius $\rho=2$ when the dual codes are antipodal. These completely regular codes are extensions of linear completely regular codes with covering radius 1, which we also classify. For $\rho=2$, we give a list of all such codes known to us. This also gives the characterization of two weight linear antipodal codes. Finally, for a class of completely regular codes with covering radius $\rho=2$ and antipodal dual, some interesting properties on self-duality and lifted codes are pointed out.

We obtain finite and asymptotic Gilbert-Varshamov type bounds for linear codes over finite chain rings with various weights.