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The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over Fq. Here b ≥ 2 is an arbitrary prescribed positive integer and Fq is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n−1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q. [ABSTRACT FROM AUTHOR]

Let $ \mathbb K$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $ \mathbb K$ , containing three points, $P_{1},P_{2}$ , and $P_{3}$ , such that $nP_{1}+P_{2}$ , $nP_{2}+P_{3}$ , and $nP_{3}+P_{1}$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_{1},P_{2},P_{3}\}$ , as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points $P_{1},P_{2}$ , and $P_{3}$. When $ \mathbb K=\overline { \mathbb F}_{q}$ , this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound. [ABSTRACT FROM AUTHOR]

Convolutional Goppa codes (CGC) were defined in Appl. Algebra Eng. Comm. Comput., vol. 15, pp. 51-61, 2004 and IEEE Trans. Inf. Theory, vol. 52, 340-344, 2006. In this paper, we prove that every convolutional code is a CGC defined over a smooth curve over \BBF q(z) and we give an explicit construction of convolutional codes as CGC over the projective line \BBP \BBF q(z)^{1}. We characterize which convolutional codes are defined by a complete linear system over curves of genus 0, 1, and over hyperelliptic curves. We apply these results to provide detailed constructions of some linear block codes as Goppa codes. [ABSTRACT FROM AUTHOR]

Multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions, which have multiple poles and/or zeros on an algebraic curve. Thus, they are more general than one-point codes, which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the Berlekamp–Massey–Sakata algorithm. We present a fast method for decoding multipoint codes from a plane curve, particularly a Hermitian curve. Our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the N-dimensional affine space \bf Fq^{N} over a finite field \bf Fq, so that those algebraic geometry codes can be decoded efficiently if the dimension N of the affine space, including the defining curve is small. [ABSTRACT FROM PUBLISHER]

Compressed sensing is a sampling technique which provides a fundamentally new approach to data acquisition. Comparing with traditional methods, compressed sensing makes full use of sparsity so that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. Inspired by algebraic geometry codes, we introduce a new deterministic construction via algebraic curves over finite fields, which is a natural generalization of DeVore's construction using polynomials over finite fields. The diversity of algebraic curves provides numerous choices for sensing matrices. By choosing appropriate curves, we are able to construct binary sensing matrices which are superior to Devore's ones. We hope this connection between algebraic geometry and compressed sensing will provide a new point of view and stimulate further research in both areas. [ABSTRACT FROM PUBLISHER]

In this correspondence, we prove that the distributions of the two-dimensional Kloosterman sums are completely determined by the distributions of the (one-dimensional) Kloosterman sums, and, by a famous result of Lauchad and Wolfman, we can give all the values of the two-dimensional Kloosterman sums. Furthermore, using the identities on one-dimensional Kloosterman sums found by Tor Helleseth and Victor Zinoviev, Hollmann and Xiang, etc., we present some new identities on Kloosterman sums in this note. Meanwhile, we also provide a new approach to producing new Kloosterman sums' identities. [ABSTRACT FROM AUTHOR]