Weight Spectrum of Quasi-Perfect Binary Codes with Distance 4

We consider the weight spectrum of a class of quasi-perfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).Comment: 5 pages, 11 references, 2 tables; some explanations and detail are adde

A single-cell Arabidopsis root atlas reveals developmental trajectories in wild-type and cell identity mutants

In all multicellular organisms, transcriptional networks orchestrate organ development. The Arabidopsis root, with its simple structure and indeterminate growth, is an ideal model for investigating the spatiotemporal transcriptional signatures underlying developmental trajectories. To map gene expression dynamics across root cell types and developmental time, we built a comprehensive, organ-scale atlas at single-cell resolution. In addition to estimating developmental progressions in pseudotime, we employed the mathematical concept of optimal transport to infer developmental trajectories and identify their underlying regulators. To demonstrate the utility of the atlas to interpret new datasets, we profiled mutants for two key transcriptional regulators at single-cell resolution, shortroot and scarecrow. We report transcriptomic and in vivo evidence for tissue trans-differentiation underlying a mixed cell identity phenotype in scarecrow. Our results support the atlas as a rich community resource for unraveling the transcriptional programs that specify and maintain cell identity to regulate spatiotemporal organ development

Comparisons between Chemical Mapping and Binding to Isoenergetic Oligonucleotide Microarrays Reveal Unexpected Patterns of Binding to the Bacillus subtilis RNase P RNA Specificity Domain†

ABSTRACT: Microarrays with isoenergetic pentamer and hexamer 20-O-methyl oligonucleotide probes with LNA (locked nucleic acid) and 2,6-diaminopurine substitutions were used to probe the binding sites on theRNase P RNA specificity domain of Bacillus subtilis. Unexpected binding patterns were revealed. Because of their enhanced binding free energies, isoenergetic probes can break short duplexes, merge adjacent loops, and/or induce refolding. This suggests new approaches to the rational design of short oligonucleotide therapeutics but limits the utility of microarrays for providing constraints for RNA structure determination. The microarray results are compared to results from chemical mapping experiments, which do provide constraints. Results from both types of experiments indicate that the RNase P RNA folds similarly in 1MNaþ and 10 mMMg2þ. Binding of RNA to RNA is important for many natural func-tions, includingproteinsynthesis (1,2), translationregulation (3,4), gene silencing (5, 6), metabolic regulation (7), RNAmodification (8, 9), etc. (10-13). Binding of oligonucleotides toRNAs is impor-tant for therapeutic approaches, such as siRNA, ribozymes, and antisense therapy (14, 15).Much remains to bediscovered, however, of the rules for predicting binding sites andpotential therapeutics