One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

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We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions.

Arsenic (As), naturally occurring metalloid and a potential hazardous material, is found in low concentrations in the environment and emerges from natural sources and anthropogenic activities. The presence of As in ground water, which is used for irrigation, is a matter of great concern since it affects crop productivity and contaminates food chain. In plants, As alters various metabolic pathways in cells including the interaction of substrates/enzymes with the sulfhydryl groups of proteins and the replacement of phosphate in ATP for energy. In addition, As stimulates the generation of free radicals and reactive oxygen species (ROS), resulting in oxidative stress. Glutathione S-transferases (GSTs) quench reactive molecules with the addition of glutathione (GSH) and protect the cell from oxidative damage. GSTs are a multigene family of isozymes, known to catalyze the conjugation of GSH to miscellany of electrophilic and hydrophobic substrates. GSTs have been reported to be associated with plant developmental processes and are responsive to multitude of stressors. In past, several studies suggested involvement of plant GST gene family in As response due to the requirement of sulfur and GSH in the detoxification of this toxic metalloid. This review provides updated information about the role of GSTs in abiotic and biotic stresses with an emphasis on As uptake, metabolism, and detoxification in plants. Further, the genetic manipulations that helped in enhancing the understanding of the function of GSTs in abiotic stress response and heavy metal detoxification has been reviewed.