11th ICTS Abstracts

This talk presents algebraic properties of prime, even, odd, primitive Eisenstein integers, and the quotient ring of Eisenstein integers. Some results are used as algebraic tools for applications to signal constellations.

Next, constructions of signal constellations over quotient rings of Eisenstein integers, equipped with Euclidean and hexagonal distances, generalizing those over Eisenstein integer fields, are discussed. By set partitioning, a quotient ring of Eisenstein integers is divided into equal-sized subsets.

In Eisenstein integer fields of prime size, partitioning is not feasible due to structural limitations. In our setup, a quotient ring is partitioned based on additive subgroups in such a way that the minimum distances within each subgroup are larger than in the original set. This technique facilitates multilevel coding and enhances the signal constellation's efficiency.

In this talk, we present a connection between three unseemingly related distance bounds for quasi-twisted (QT) codes, i.e., the Lally bound, the Jensen bound, and the spectral bound.

We first extend the first two bounds to the QT codes setting, then establish a comparison of the three bounds. Moreover, numerical comparisons illustrating how all these three bounds behave over a large number of randomly chosen QT codes will also be presented.

Lastly, we provide recent developments on this area of study.

We study a non-unital and non-commutative ring Sm(R), called the ring of ordered sum over a ring R. We obtain linear codes over Sm(F2), also known as Sm-codes, where F2 is the binary field.

The algebraic structure of Sm-codes, particularly their residue and torsion codes, will be explored. Moreover, a generalized notion of quasi self-dual codes and even quasi self-dual codes will be introduced.

Finally, we give results on the weight enumerators of these codes and construct S3-codes in short lengths using the residue and torsion codes.