Systematic MDS erasure codes based on vandermonde matrices

An increasing number of applications in computer communications uses erasure codes to cope with packet losses. Systematic maximum-distance separable (MDS) codes are often the best adapted codes. This letter introduces new systematic MDS erasure codes constructed from two Vandermonde matrices. These codes have lower coding and decoding complexities than the others systematic MDS erasure codes

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

Quantum generalized Reed-Solomon codes: Unified framework for quantum MDS codes

We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of this type with the same parameters also exists. Thus as far as is known at present, they are the most important family of quantum MDS codes.Comment: 9 pages, no figure

A population-based study on myelodysplastic syndromes in the Lazio Region (Italy), medical miscoding and 11-year mortality follow-up. The Gruppo Romano-Laziale Mielodisplasie experience of retrospective multicentric registry

Data on Myelodysplastic Syndromes (MDS) are difficult to collect by cancer registries because of the lack of reporting and the use of different classifications of the disease. In the Lazio Region, data from patients with a confirmed diagnosis of MDS, treated by a hematology center, have been collected since 2002 by the Gruppo Romano-Laziale Mielodisplasie (GROM-L) registry, the second MDS registry existing in Italy. This study aimed at evaluating MDS medical miscoding during hospitalizations, and patients' survival. For these purposes, we selected 644 MDS patients enrolled in the GROM-L registry. This cohort was linked with two regional health information systems: the Hospital Information System (HIS) and the Mortality Information System (MIS) in the 2002-2012 period. Of the 442 patients who were hospitalized at least once during the study period, 92% had up to 12 hospitalizations. 28.5% of patients had no hospitalization episodes scored like MDS, code 238.7 of the International Classification of Disease, Ninth Revision, Clinical Modification (ICD-9-CM). The rate of death during a median follow-up of 46 months (range 0.9-130) was 45.5%. Acute myeloid leukemia (AML) was the first cause of mortality, interestingly a relevant portion of deaths is due to cerebro-cardiovascular events and second tumors. This study highlights that MDS diagnosis and treatment, which require considerable healthcare resources, tend to be under-documented in the HIS archive. Thus we need to improve the HIS to better identify information on MDS hospitalizations and outcome. Moreover, we underline the importance of comorbidity in MDS patients' survival