In this work we investigate unions of lifted MRD codes of a fixed dimension and minimum distance and derive an explicit formula for the cardinality of such codes. This will then imply a lower bound on the cardinality of constant dimension codes. We will first repeat some known results needed in this section. In Section 2 we will explain the construction and derive the formula for the cardinalit...

Spread codes and orbit codes are special families of constant dimension subspace codes. These codes have been wellstudied for their error correction capability and transmission rate, but the question of how to encode messages has not been investigated. In this work we show how the message space can be chosen for a given code and how message enand decoding can be done.

We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, we construct a ...

Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Plücker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words. 1. Introduction. In network coding one is looking at the transmission of information through a directed graph with possibly several senders and several receivers [1]. One c...

We present a family of constant–dimension codes for random linear network coding called spread codes. This is a family of optimal codes with maximum minimum distance. A spread code is constructed starting from the algebra defined by the companion matrix of an irreducible polynomial. We give a minimum distance decoding algorithm that is particularly efficient when the dimension of the codewords ...

Rank metric codes can either be used as such for error correction in data storage equipments, or be lifted into constant-dimension codes (CDCs) and thus be used for error correction in random network coding. This paper investigates the decoder error probability (DEP) of rank metric codes and CDCs. We first study the DEP of rank metric codes using a bounded rank distance decoder. We derive asymp...

Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been wellstudied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this work we show how a message set of consecutive integers can be encoded and retrieved for these two ...

Distributed storage systems based on equidistant constant dimension codes are presented. These equidistant codes are based on the Plücker embedding, which is essential in the repair and the reconstruction algorithms. These systems posses several useful properties such as high failure resilience, minimum bandwidth, low storage, simple algebraic repair and reconstruction algorithms, good locality...