On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4

A subset $S$ of $\{0,1,...,2t-1\}^n$ is called a $t$-fold MDS code if every line in each of $n$ base directions contains exactly $t$ elements of $S$. The adjacency graph of a $t$-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of $t$-fold MDS codes of smaller lengths. In the case $t=2$, the theory has the following application. The union of two disjoint $(n,4^{n-1},2)$ MDS codes in $\{0,1,2,3\}^n$ is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation as a test for reducibility of $n$-quasigroups of order 4. Keywords: MDS codes, n-quasigroups, decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Mat

Application of Constacyclic codes to Quantum MDS Codes

Quantum maximal-distance-separable (MDS) codes form an important class of quantum codes. To get $q$-ary quantum MDS codes, it suffices to find linear MDS codes $C$ over $\mathbb{F}_{q^2}$ satisfying $C^{\perp_H}\subseteq C$ by the Hermitian construction and the quantum Singleton bound. If $C^{\perp_{H}}\subseteq C$, we say that $C$ is a dual-containing code. Many new quantum MDS codes with relatively large minimum distance have been produced by constructing dual-containing constacyclic MDS codes (see \cite{Guardia11}, \cite{Kai13}, \cite{Kai14}). These works motivate us to make a careful study on the existence condition for nontrivial dual-containing constacyclic codes. This would help us to avoid unnecessary attempts and provide effective ideas in order to construct dual-containing codes. Several classes of dual-containing MDS constacyclic codes are constructed and their parameters are computed. Consequently, new quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have parameters better than the ones available in the literature.Comment: 16 page

MDS matrices over small fields: A proof of the GM-MDS conjecture

An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over any field, as well as sufficient over very large fields by a probabilistic argument. Dau et al. (ISIT 2014) conjectured that the MDS condition is sufficient over small fields as well, where the construction of the matrix is algebraic instead of probabilistic. This is known as the GM-MDS conjecture. Concretely, if a $k \times n$ zero pattern satisfies the MDS condition, then they conjecture that there exists an MDS matrix with this zero pattern over any field of size $|\mathbb{F}| \ge n+k-1$. In recent years, this conjecture was proven in several special cases. In this work, we resolve the conjecture

Myelodysplastic syndromes: Aspects of current medical care and economic considerations in Germany

Myelodysplastic syndromes (MDS) are a heterogeneous group of diseases mainly affecting older people. The use of an increasing number of therapeutic options depends on a systematic risk stratification of the patients. A high percentage of MDS patients need blood transfusions as supportive care, which influence quality of life and cause a great part of the costs generated by MDS therapy. In this article which is based on a workshop about the burden of MDS held in October 2006 in Munich, MDS is discussed with regard to different aspects: current therapies, transfusion medicine, geriatrics, quality of life, and health economic aspects