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In this paper we investigate known Singleton-like bounds in the Lee metric and characterize optimal codes, which turn out to be very few. We then focus on Plotkin-like bounds in the Lee metric and present a new bound that extends and refines a previously known, and out-performs it in the case of non-free codes. We then compute the density of optimal codes with regard to the new bound. Finally we fill a gap in the characterization of Lee-equidistant codes.

We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise asymptotic estimates on the density functions of several classes of codes that are extremal with respect to minimum distance, covering radius, and maximality. The techniques developed in this paper apply to various distance functions, including the Hamming and the rank metric distances. Applications of our results show that, unlike the F q m -linear MRD codes, the F q -linear MRD codes are not dense in the family of codes of the same dimension. More precisely, we show that the density of F q -linear MRD codes in F q n × m in the set of all matrix codes of the same dimension is asymptotically at most 1/2, both as q → + ∞ and as m → + ∞ . We also prove that MDS and F q m -linear MRD codes are dense in the family of maximal codes. Although there does not exist a direct analogue of the redundancy bound for the covering radius of F q -linear rank metric codes, we show that a similar bound is satisfied by a uniformly random matrix code with high probability. In particular, we prove that codes meeting this bound are dense. Finally, we compute the average weight distribution of linear codes in the rank metric, and other parameters that generalize the total weight of a linear code.

We introduce two new infinite families of APN functions, one on fields of order 22k for k not divisible by 2, and the other on fields of order 23k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second family's polynomials have three terms.

We develop a theory of Gröbner bases over Galois rings, following the usual formulation for Gröbner bases over finite fields. Our treatment includes a division algorithm, a characterization of Gröbner bases, and an extension of Buchberger’s algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M= {(a, b) :aS≡b modxr} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (Σ, Ω) ∈M is sought satisfying certain conditions, and such a solution can be found in a Gröbner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.

We investigate a class of Lee-metric alternant codes with symbols in Z p n , establishing a lower bound on the minimum Lee distance where certain restrictions are placed on the code parameters. Corresponding to this bound we have devised two decoding algorithms. The first operates entirely over a Galois ring, while the second must be implemented over the associated residue field. The algorithms proceed by finding a Grobner basis of the module M of solutions to a key equation. We obtain a necessary characterisation of the solution module by solving iteratively a linear sequence over a Galois ring and show that the particular solution sought by the decoder is minimal in M . Hence the required solution can be found in an appropriate Grobner basis of M .