Codes and Designs Related to Lifted MRD Codes

Lifted maximum rank distance (MRD) codes, which are constant dimension codes, are considered. It is shown that a lifted MRD code can be represented in such a way that it forms a block design known as a transversal design. A slightly different representation of this design makes it similar to a $q-$analog of a transversal design. The structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted MRD code. Codes which attain these bounds are constructed. These codes are the largest known codes for the given parameters. These transversal designs can be also used to derive a new family of linear codes in the Hamming space. Bounds on the minimum distance and the dimension of such codes are given.Comment: Submitted to IEEE Transactions on Information Theory. The material in this paper was presented in part in the 2011 IEEE International Symposium on Information Theory, Saint Petersburg, Russia, August 201

Message Encoding for Spread and Orbit Codes

Spread codes and orbit codes are special families of constant dimension subspace codes. These codes have been well-studied 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 en- and decoding can be done.Comment: Submitted to IEEE International Symposium on Information Theory 201

Optimal Binary Locally Repairable Codes via Anticodes

This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound

Subspaces intersecting in at most a point

We improve on the lower bound of the maximum number of planes in \operatorname{PG}(8,q)\cong\F_q^{9} pairwise intersecting in at most a point. In terms of constant dimension codes this leads to $A_q(9,4;3)\ge q^{12}+ 2q^8+2q^7+q^6+2q^5+2q^4-2q^2-2q+1$. This result is obtained via a more general construction strategy, which also yields other improvements.Comment: 4 page