New constructions of optimal (r,δ)(r,\delta)-LRCs via good polynomials

Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal $(r,\delta)$-LRCs based on polynomial evaluation. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of $\mathbb{F}_q$. In this paper, we extend the aforementioned constructions of RS-like LRCs and proposed new constructions of $(r,\delta)$-LRCs whose code length can be larger. These new $(r,\delta)$-LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance, that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combing these constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal $(r,\delta)$-LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like $(r,\delta)$-LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal $(r,\delta)$-LRCs with length $n=q-1+\delta$ for any positive integers $r,\delta\geq 2$ and $(r+\delta-1)\mid (q-1)$. When $\delta$ is proportional to $q$, it is asymptotically longer than that constructed via elliptic curves whose length is at most $q+2\sqrt{q}$. Besides, it allows more flexibility on the values of $r$ and $\delta$

POWERPLAY: Training an Increasingly General Problem Solver by Continually Searching for the Simplest Still Unsolvable Problem

Most of computer science focuses on automatically solving given computational problems. I focus on automatically inventing or discovering problems in a way inspired by the playful behavior of animals and humans, to train a more and more general problem solver from scratch in an unsupervised fashion. Consider the infinite set of all computable descriptions of tasks with possibly computable solutions. The novel algorithmic framework POWERPLAY (2011) continually searches the space of possible pairs of new tasks and modifications of the current problem solver, until it finds a more powerful problem solver that provably solves all previously learned tasks plus the new one, while the unmodified predecessor does not. Wow-effects are achieved by continually making previously learned skills more efficient such that they require less time and space. New skills may (partially) re-use previously learned skills. POWERPLAY's search orders candidate pairs of tasks and solver modifications by their conditional computational (time & space) complexity, given the stored experience so far. The new task and its corresponding task-solving skill are those first found and validated. The computational costs of validating new tasks need not grow with task repertoire size. POWERPLAY's ongoing search for novelty keeps breaking the generalization abilities of its present solver. This is related to Goedel's sequence of increasingly powerful formal theories based on adding formerly unprovable statements to the axioms without affecting previously provable theorems. The continually increasing repertoire of problem solving procedures can be exploited by a parallel search for solutions to additional externally posed tasks. POWERPLAY may be viewed as a greedy but practical implementation of basic principles of creativity. A first experimental analysis can be found in separate papers [53,54].Comment: 21 pages, additional connections to previous work, references to first experiments with POWERPLA

Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound

A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois