Binary and Ternary Quasi-perfect Codes with Small Dimensions

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First we give a list of infinite families of QP codes which includes all binary, ternary and quaternary codes known to is. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.Comment: 4 page

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems

We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath. We propose a phenomenological model for the resulting short-time dynamics that includes pair-creation, hopping, braiding, and fusion of anyons. By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces. To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting. We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range between 13% to 25%. To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.Comment: 34 pages, 9 figures; v2 matches the journal version and corrects a misstatement about the detailed balance condition of our Metropolis simulations. All conclusions from v1 are unaffected by this correctio

Algebraic Hybrid Satellite-Terrestrial Space-Time Codes for Digital Broadcasting in SFN

Lately, different methods for broadcasting future digital TV in a single frequency network (SFN) have been under an intensive study. To improve the transmission to also cover suburban and rural areas, a hybrid scheme may be used. In hybrid transmission, the signal is transmitted both from a satellite and from a terrestrial site. In 2008, Y. Nasser et al. proposed to use a double layer 3D space-time (ST) code in the hybrid 4 x 2 MIMO transmission of digital TV. In this paper, alternative codes with simpler structure are proposed for the 4 x 2 hybrid system, and new codes are constructed for the 3 x 2 system. The performance of the proposed codes is analyzed through computer simulations, showing a significant improvement over simple repetition schemes. The proposed codes prove in addition to be very robust in the presence of power imbalance between the two sites.Comment: 14 pages, 2 figures, submitted to ISIT 201

An Adaptive Entanglement Distillation Scheme Using Quantum Low Density Parity Check Codes

Quantum low density parity check (QLDPC) codes are useful primitives for quantum information processing because they can be encoded and decoded efficiently. Besides, the error correcting capability of a few QLDPC codes exceeds the quantum Gilbert-Varshamov bound. Here, we report a numerical performance analysis of an adaptive entanglement distillation scheme using QLDPC codes. In particular, we find that the expected yield of our adaptive distillation scheme to combat depolarization errors exceed that of Leung and Shor whenever the error probability is less than about 0.07 or greater than about 0.28. This finding illustrates the effectiveness of using QLDPC codes in entanglement distillation.Comment: 12 pages, 6 figure