Abstract We introduce the concept of a sum–rank saturating system and outline its correspondence to covering properties of a sum–rank metric code. We consider the problem of determining the shortest length of a sum–rank- $$\rho $$ ρ -saturating system of a fixed dimension, which is equivalent to the covering problem in the sum–rank metric. We obtain upper and lower bounds on this quantity. We also give constructions of saturating systems arising from geometrical structures.

Abstract. Strong blocking sets and their counterparts, minimal codes, have attracted much attention in the past few years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.

We develop three approaches of combinatorial flavor to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon--Füredi theorem and the combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. By applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.

Let$C$be a binary self-dual code with an automorphism$g$of order$2p$, where$p$is an odd prime, such that$g^{p}$is a fixed point free involution. If$C$is extremal of length a multiple of 24, all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code$C$with coding theoretical ones of the subcode$C(g^{p})$which consists of the set of fixed points of$g^{p}$, we prove that$C$is a projective$ {\BBF }_{2}\langle g \rangle $-module if and only if a natural projection of$C(g^{p})$is a self-dual code. We then discuss easy-to-handle criteria to decide if$C$is projective or not. As an application, we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.

The existence of an extremal code of length 72 is a long-standing open problem. LetCbe a putative extremal code of length 72 and suppose thatChas an automorphismgof order 6. We show thatC, as an \BBF2〈g〉 -module, is the direct sum of two modules; it is easy to determine one of them, while the other one has a very restrictive structure. We use this fact to do an exhaustive search and we do not find an extremal code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.