Problems on q-Analogs in Coding Theory

The interest in $q$-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author

Equidistant Codes in the Grassmannian

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Pl\"{u}cker embedding, for 1-intersecting codes of $k$-dimensional subspaces over \F_q^n, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2}$ over \F_q, rank $n-1$, and rank distance $n-1$.Comment: 16 page

Network Coding for Multi-Resolution Multicast

Multi-resolution codes enable multicast at different rates to different receivers, a setup that is often desirable for graphics or video streaming. We propose a simple, distributed, two-stage message passing algorithm to generate network codes for single-source multicast of multi-resolution codes. The goal of this "pushback algorithm" is to maximize the total rate achieved by all receivers, while guaranteeing decodability of the base layer at each receiver. By conducting pushback and code generation stages, this algorithm takes advantage of inter-layer as well as intra-layer coding. Numerical simulations show that in terms of total rate achieved, the pushback algorithm outperforms routing and intra-layer coding schemes, even with codeword sizes as small as 10 bits. In addition, the performance gap widens as the number of receivers and the number of nodes in the network increases. We also observe that naiive inter-layer coding schemes may perform worse than intra-layer schemes under certain network conditions.Comment: 9 pages, 16 figures, submitted to IEEE INFOCOM 201

Covering of Subspaces by Subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph \cG_q(n,k), $k \geq r$, are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, $q$-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for $q=2$ with $r=2$ or $r=3$. We discuss the density for some of these coverings. Tables for the best known coverings, for $q=2$ and $5 \leq n \leq 10$, are presented. We present some questions concerning possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352

Entanglement-assisted quantum low-density parity-check codes

This paper develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes (EAQECCs) with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error correction performance, high rates, and low decoding complexity. The proposed method produces infinitely many new codes with a wide variety of parameters and entanglement requirements. Our framework encompasses various codes including the previously known entanglement-assisted quantum LDPC codes having the best error correction performance and many new codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.Comment: 20 pages, 5 figures. Final version appearing in Physical Review