Locally recoverable (LRC) codes provide ways of recovering erased coordinates of the codeword without having to access each of the remaining coordinates. A subfamily of LRC codes with hierarchical locality (H-LRC codes) provides added flexibility to the construction by introducing several tiers of recoverability for correcting different numbers of erasures. We present a general construction of codes with 2-level hierarchical locality from maps between algebraic curves and specialize it to several code families obtained from quotients of curves by a subgroup of the automorphism group, including rational, elliptic, Kummer, and Artin-Schreier curves. We further address the question of H-LRC codes with availability, and suggest a general construction of such codes from fiber products of curves. Detailed calculations of parameters for H-LRC codes with availability are performed for Reed-Solomon- and Hermitian-like code families. Finally, we construct asymptotically good families of H-LRC codes from curves related to the Garcia-Stichtenoth tower.

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) of other symbols of the codeword. In this paper, we introduce a construction of LRC codes on algebraic curves, extending a recent construction of the Reed-Solomon like codes with locality. We treat the following situations: local recovery of a single erasure, local recovery of multiple erasures, and codes with several disjoint recovery sets for every coordinate (the availability problem). For each of these three problems we describe a general construction of codes on curves and construct several families of LRC codes. We also describe a construction of codes with availability that relies on automorphism groups of curves. We also consider the asymptotic problem for the parameters of the LRC codes on curves. We show that the codes obtained from asymptotically maximal curves (for instance, Garcia-Stichtenoth towers) improve upon the asymptotic versions of the Gilbert-Varshamov bound for LRC codes.

A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, vol. 60, no. 8, 2014, pp. 4661-4676). In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia-Stichtenoth towers. The local recovery procedure is performed by polynomial interpolation over r coordinates of the codevector. We also obtain a family of Hermitian codes with two disjoint recovering sets for every symbol of the codeword.

In this note we show that one can ameliorate existing lower bounds for parameters of authentication codes using algebraic-geometry codes.

The notion of higher (or generalized) weights of codes is just as natural as that of the classical Hamming weight. The authors adopt the geometric point of view and always treat the q-ary case. Some results and proofs being new, the main goal is to present a clear picture of what is known on the subject.

Studies a class of projective systems and linear codes corresponding to Hermitian varieties over finite fields. The weight hierarchy, also known as the set of generalized Hamming weights, of the code is calculated. The higher weight distribution is also found.>

A decoding algorithm for algebraic-geometric codes arising from arbitrary algebraic curves is presented. This algorithm corrects any number of errors up to ((d-g-1)/2), where d is the designed distance of the code and g is the genus of the curve. The complexity of decoding equals sigma (n/sup 3/) where n is the length of the code. Also presented is a modification of this algorithm, which in the case of elliptic and hyperelliptic curves is able to correct ((d-1)/2) errors. It is shown that for some codes based on plane curves the modified decoding algorithm corrects approximately d/2-g/4 errors. Asymptotically good q-ary codes with a polynomial construction and a polynomial decoding algorithm (for q>or=361 on some segment their parameters are better than the Gilbert-Varshamov bound) are obtained. A family of asymptotically good binary codes with polynomial construction and polynomial decoding is also obtained, whose parameters are better than the Blokh-Zyablov bound on the whole interval 0>

It is proved that for algebraic-geometric codes on a curve over F/sub q/ for q>or=37 or on a curve of sufficiently large genus over F/sub q/ for q>or=16 there exists a polynomial decoding algorithm up to (d*-1)/2 errors, d* being the designed minimum distance.>

In this paper we studyq-ary codes arising from modular curvesX_{0}(11l)over GF(p^{2})and some binary codes attached to them. All these codes have a polynomial construction and have "good" asymptotic parameters: for certain values of the parameters theq-ary codes withq = p^{2} \geq 49are better than the Varshamov-Gilbert bound, and the binary codes are better than the Ziablov bounds.