Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.Comment: 17 pages, 4 figure

Algebraic curves and applications to coding theory.

by Yan Cho Hung.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 122-124).Abstract also in Chinese.Chapter 1 --- Complex algebraic curves --- p.6Chapter 1.1 --- Foundations --- p.6Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9Chapter 1.1.3 --- Complex projective curves in P2 --- p.11Chapter 1.1.4 --- Affine and projective curves --- p.13Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16Chapter 1.2.1 --- Intersection multiplicity --- p.16Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18Chapter 1.2.3 --- Cubic curves --- p.21Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23Chapter 1.4 --- Riemann surfaces --- p.26Chapter 1.4.1 --- Weierstrass &-function --- p.26Chapter 1.4.2 --- Riemann surfaces and examples --- p.27Chapter 1.5 --- Differentials on Riemann surfaces --- p.28Chapter 1.5.1 --- Holomorphic differentials --- p.28Chapter 1.5.2 --- Abel's Theorem for tori --- p.31Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32Chapter 1.6 --- Singular curves --- p.36Chapter 1.6.1 --- Resolution of singularities --- p.37Chapter 1.6.2 --- The topology of singular curves --- p.45Chapter 2 --- Coding theory --- p.48Chapter 2.1 --- An introduction to codes --- p.48Chapter 2.1.1 --- Efficient noiseless coding --- p.51Chapter 2.1.2 --- The main coding theory problem --- p.56Chapter 2.2 --- Linear codes --- p.58Chapter 2.2.1 --- Syndrome decoding --- p.63Chapter 2.2.2 --- Equivalence of codes --- p.65Chapter 2.2.3 --- An introduction to cyclic codes --- p.67Chapter 2.3 --- Special linear codes --- p.71Chapter 2.3.1 --- Hamming codes --- p.71Chapter 2.3.2 --- Simplex codes --- p.72Chapter 2.3.3 --- Reed-Muller codes --- p.73Chapter 2.3.4 --- BCH codes --- p.75Chapter 2.4 --- Bounds on codes --- p.77Chapter 2.4.1 --- Spheres in Zn --- p.77Chapter 2.4.2 --- Perfect codes --- p.78Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85Chapter 3.1 --- Algebraic curves over finite fields --- p.85Chapter 3.1.1 --- Affine varieties --- p.85Chapter 3.1.2 --- Projective varieties --- p.37Chapter 3.1.3 --- Morphisms --- p.89Chapter 3.1.4 --- Rational maps --- p.91Chapter 3.1.5 --- Non-singular varieties --- p.92Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93Chapter 3.2 --- Goppa codes --- p.96Chapter 3.2.1 --- Elementary Goppa codes --- p.96Chapter 3.2.2 --- The affine and projective lines --- p.98Chapter 3.2.3 --- Goppa codes on the projective line --- p.102Chapter 3.2.4 --- Differentials and divisors --- p.105Chapter 3.2.5 --- Algebraic geometric codes --- p.112Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116Bibliograph

A Geometric View of the Service Rates of Codes Problem and its Application to the Service Rate of the First Order Reed-Muller Codes

Service rate is an important, recently introduced, performance metric associated with distributed coded storage systems. Among other interpretations, it measures the number of users that can be simultaneously served by the storage system. We introduce a geometric approach to address this problem. One of the most significant advantages of this approach over the existing approaches is that it allows one to derive bounds on the service rate of a code without explicitly knowing the list of all possible recovery sets. To illustrate the power of our geometric approach, we derive upper bounds on the service rates of the first order Reed-Muller codes and simplex codes. Then, we show how these upper bounds can be achieved. Furthermore, utilizing the proposed geometric technique, we show that given the service rate region of a code, a lower bound on the minimum distance of the code can be obtained

Subfield subcodes of projective Reed-Muller codes

Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we are able to generalize the necessary tools to deal with this case as well: we obtain a universal Gr\"obner basis for the vanishing ideal of the set of standard representatives of the projective space and we are able to reduce any monomial with respect to this Gr\"obner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we are able to obtain long linear codes with good parameters over a small finite field

Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed-Solomon codes

We study the subfield subcodes of projective Reed-Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature