chapter, introduction, thesis, problem, data, storage, distributed, systems, stored, servers, operate, independently, us, example, system, consisting, nservers, assume, user, server, multiple, users, consider, scenario, situation, requests, satised, redundantly, situations, described, preceding, paragraph, formalized, locally, repairable, codes, lrcs, local, symbols, process, recovering, erased, symbol, setup, encoded, failed, mapped, literature, standard, notations, classical, coding, theory, drefer, dimension, minimum, distance, code, dened, nite, eld, respectively, notion, locality, recovered, introduced, yekhanin, et, al, performed, depth, relationships, redundancy, special, main, results, rdenotes, information, repaired, rother, recognizing, unsuccessful, rsymbols, wang, zhang, introduce, availability, tdisjoint, derive, bound, linearcodes, re, result, derived, dimakis, non, tamo, barg, deriving, symbollocality, obtain, generalized, mazumdar, cooperative, recover, disjoint, ldenotes, erasures, zdenotes, maximum, required, contribution, introducing, generalization, named, multi, rcs, dierent, construction, collection, repetitions, allowed, mlrcs, existing, bounds, organization, introduces, models, address, section, brief, overview, necessary, notions, unfamiliar, concepts, relevant, generic, established, model, suggests, brie, comment, research, suggest, sequence, sequentially, advanced, sections, remaining, encouraged, generalizations, abbreviated, clrcs, preliminaries, avia, brock, represent, kbit, vector, fk, ideally, medium, errors, intended, practice, mediums, ipped, transmission, issue, error, correcting, simply, transmitted, vectors, particular, linear, codeword, matrix, called, generator, matrixof, fol, lowing, wil, cal, encoding, avias, encodedto, simplex, xis, using, generate, immediately, rst, yare, viz, coincidence, encode, directly, codewords, referred, systematic, normally, parameters, within, brackets, d], represented, kis, usually, nmatrix, linearly, independent, hamming, coordinates, dier, instance, formally, jfi, igj, denotes, aand, ja, denoted, distances, nonlinear, cconsisting, combination, ccan, fy, combinations, relationship, captured, richard, collom, singleton, binary, yis, reconstruct, denote, th, indices, possibilities, equations, groupsof, union, intersecting, consideration, needed, thus, yhas, total, denition, satises, properties, trepair, exist, tsets, function, indexed, [k, [n, ]nf, ig, [t, encodes, ysuch, obvious, showing, listed, de, property, contains, fth, discussion, intersect, shown, studied, codewordy, functions, zother, js, sj, demonstrate, minor, amendment, proof, theorem, tighter, perform, detailed, basics, tnon, tuple, oftsymbols, elements, necessarily, distinct, reconstructed, ]n, tg, discuss, yencodes, argue, discussing, tuples, allowing, repetition, claim, ninth, tenth, mentioned, convince, rt, tand, appeared, derives, later, optimize, parameter, require, letcbe, cby, constructing, sub, xed, smaller, min, dmin, inequality, analysis, based, techniques, pro, ject, onto, iteration, jection, popular, correspond, construct, repeated, dene, tothe, ij, [j, jas, inthe, aj, jj, relations, jin, algorithm, input, jc, fij, coordinate, tindices, letr, jbe, index, rtcode, tsymbols, lety, fj, frequent, element, xj, maximal, subset, jand, replace, occurrences, jby, output, consists, jr, qj, truncated, factor, induction, assuming, terminates, qjc, qqp, recall, j[, plugged, inequalities, leading, noting, ji, formed, puncturing, applying, combining, nally, corollary, cmust, greater, tto, terasures, yields, optimizing, presented, estimate, optimizes, adjustment, penalty, adding, atop, derivative, respect, ignoring, operation, equating, zero, plugging, incurred, pmlrc, plrc, kt, restrictive, tinto, te, dte, comparison, lrepresents, zrepresents, translate, pclrc, kand, noticed, considers, mandatorily, reasonable, extension, balancing, requirements, necessitate, haveconcentrated, adapting, concept, acknowledgments, express, sincere, gratitude, european, commission, funding, participation, nordsecmob, program, dr, vitaly, skachek, guidance, hui, discussions, appreciate, support, baba, mamu, shubu