Remarks on Bounds for Quantum Codes

We present some results that show that bounds from classical coding theory still work in many cases of quantum coding theory

Correction of Data and Syndrome Errors by Stabilizer Codes

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes and provide several CSS-type code constructions of quantum DS codes.Comment: 2 figures. This is a short version of our full paper (in preparation

Asymptotically Good Quantum Codes

Using algebraic geometry codes we give a polynomial construction of quantum codes with asymptotically non-zero rate and relative distance.Comment: 15 pages, 1 figur

The Finite Basis Problem for Kiselman Monoids

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $\mathcal{K}_n$. As a consequence, we conclude that the identities of $\mathcal{K}_n$ are nonfinitely based for every $n\ge 4$ and exhibit a finite identity basis for the identities of each of the monoids $\mathcal{K}_2$ and $\mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.Comment: 16 pages, 1 table, 1 figur