Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.

Let I G be the binomial edge ideal on the generic 2 × n -Hankel matrix associated with a closed graph G on the vertex set [n]. We characterize the graphs G for which I G has maximal regularity and is Gorenstein.2010 Mathematics Subject Classification. 13H10,13P10,13D02.

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension ℵω is a strong limit cardinal and ℵω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

Assuming the existence of a λ + -hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ ++ has the tree property and 2 κ = κ +++ . If the assumption is strengthened to the existence of a θ-hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get 2 κ = θ.