Applications of Grobner Bases to Linear Codes.

Put A={\bf F}\sb{q}\lbrack x\sb1,\...,x\sb{s}\rbrack, and let I be an ideal of A. Let P\sb1,\...,P\sb{n} be all the {\bf F}\sb{q}-rational points of $V(I)$. Define a map \varphi:A/I\to{\bf F}\sbsp{q}{n} by \varphi(\tilde f)=(f(P\sb1),\...,f(P\sb{n})), where f is any preimage of $\tilde f$ under the canonical map from A to $A/I.$ Let \{\tilde f\sb{i}\vert i\in {\bf N}\} be a basis of $A/I$ as an {\bf F}\sb{q}-vector space. Define the affine variety code C and its dual C\sp\perp byC=\varphi(\langle\tilde f\sb1,\...,\tilde f\sb{m}\rangle);C\sp\perp is the orthogonal complement of C with respect to the usual inner product in {\bf F}\sbsp{q}{n}. We show that any linear code can be expressed as an affine variety code. When a code C is represented as an affine variety code, problems of decoding and finding the minimum distance of C may be expressed as questions about polynomial ideals. Using the theory of Grobner bases, along with computer programs that calculate Grobner bases, we show how to decode and find the minimum distance of any linear code

Decoding Reed Solomon and BCH codes beyond their error-correction radius: an euclidean approach

In questo lavoro viene presentato un algoritmo alternativo per il list decoding di codici Reed-Solomon e BCH basato sull'algoritmo di divisione euclidea. Fissato un numero e, e data una parola ricevuta, l'obiettivo e' quello di dare in output una lista di parole del codice che abbiano distanza al piu' e da essa. Per i codici BCH la decodifica avviene attraverso la ricerca del polinomio locatore dell'errore, un particolare polinomio che si trova nel nucleo della matrice delle sindromi e che ha tutte le radici nel campo di partenza. Utilizzando queste proprieta', e attraverso l'algoritmo di divisione euclidea, siamo in grado di elencare tutti i possibili polinomi locatori dell'errore, e quindi tutte le parole del codice che abbiano la distanza desiderata. Vengono poi analizzati gli aspetti computazionali di tale algoritmo, nel caso generale e in alcuni casi particolari. Infine vengono fatti dei confronti con gli algoritmi gia' esistenti, e vengono studiati dei bound sul numero massimo di parole che l'algoritmo restituisce

Efficient soft decoding techniques for reed-solomon codes

The main focus of this thesis is on finding efficient decoding methods for Reed-Solomon (RS) codes, i.e., algorithms with acceptable performance and affordable complexity. Three classes of decoders are considered including sphere decoding, belief propagation decoding and interpolation-based decoding. Originally proposed for finding the exact solution of least-squares problems, sphere decoding (SD) is used along with the most reliable basis (MRB) to design an efficient soft decoding algorithm for RS codes. For an (N, K ) RS code, given the received vector and the lattice of all possible transmitted vectors, we propose to look for only those lattice points that fall within a sphere centered at the received vector and also are valid codewords. To achieve this goal, we use the fact that RS codes are maximum distance separable (MDS). Therefore, we use sphere decoding in order to find tentative solutions consisting of the K most reliable code symbols that fall inside the sphere. The acceptable values for each of these symbols are selected from an ordered set of most probable transmitted symbols. Based on the MDS property, K code symbols of each tentative solution can he used to find the rest of codeword symbols. If the resulting codeword is within the search radius, it is saved as a candidate transmitted codeword. Since we first find the most reliable code symbols and for each of them we use an ordered set of most probable transmitted symbols, candidate codewords are found quickly resulting in reduced complexity. Considerable coding gains are achieved over the traditional hard decision decoders with moderate increase in complexity. Due to their simplicity and good performance when used for decoding low density parity check (LDPC) codes, iterative decoders based on belief propagation (BP) have also been considered for RS codes. However, the parity check matrix of RS codes is very dense resulting in lots of short cycles in the factor graph and consequently preventing the reliability updates (using BP) from converging to a codeword. In this thesis, we propose two BP based decoding methods. In both of them, a low density extended parity check matrix is used because of its lower number of short cycles. In the first method, the cyclic structure of RS codes is taken into account and BP algorithm is applied on different cyclically shifted versions of received reliabilities, capable of detecting different error patterns. This way, some deterministic errors can be avoided. The second method is based on information correction in BP decoding where all possible values are tested for selected bits with low reliabilities. This way, the chance of BP iterations to converge to a codeword is improved significantly. Compared to the existing iterative methods for RS codes, our proposed methods provide a very good trade-off between the performance and the complexity. We also consider interpolation based decoding of RS codes. We specifically focus on Guruswami-Sudan (GS) interpolation decoding algorithm. Using the algebraic structure of RS codes and bivariate interpolation, the GS method has shown improved error correction capability compared to the traditional hard decision decoders. Based on the GS method, a multivariate interpolation decoding method is proposed for decoding interleaved RS (IRS) codes. Using this method all the RS codewords of the interleaved scheme are decoded simultaneously. In the presence of burst errors, the proposed method has improved correction capability compared to the GS method. This method is applied for decoding IRS codes when used as outer codes in concatenated code

New algebraic decoding of the( 73,37,13) quadratic residue code

为了将InVErSE-frEE bErlEkAMP-MASSEy(IfbM)算法用于平方剩余(QuAdrATIC rESIduE,Qr)码的译码,必须对未知校正子进行计算以获得连续校正子。现有算法所得数据无法从理论上保证对于所有可纠的错误图案,均能解得与该错误图案相对应的未知校正子,因此由该算法所得的数据需借助于仿真验证,非常耗时。鉴于此,提出一种改进算法,所得数据从理论上可保证对于所有可纠的错误图案,均能得到与之相应的未知校正子。基于该改进算法,提出了(73,37,13)Qr码的代数硬判决译码算法,并对所有可纠的错误图案(共185 859 898个)进行穷举仿真测试,结果验证了译码算法的正确性。In order to utilize the inverse-free Berlekamp-Massey algorithm to decode quadratic residue codes,it is essential to compute unknown syndromes to construct the enough consecutive syndromes that are needed for the application of the IFBM algorithm.But for all the decodable error patterns,the critical data which are obtained from the previous algorithms cannot guarantee theoretically obtaining the right unknown syndromes corresponding to every error pattern.In view of this,in this paper,an improved algorithm is proposed to theoretically assure the right unknown syndromes corresponding to every decodable error pattern.Based on the modified algorithm,an algebraic decoding algorithm of the( 73,37,13) quadratic residue code is depicted,and exhaustive tests of all decodable error patterns whose exact number is 185859898 are conducted successfully.国家自然科学基金(60972053)~

Contributions to folded reed-solomon codes for burst error correction

Ph.DDOCTOR OF PHILOSOPH

A Combinatorial Commutative Algebra Approach to Complete Decoding

Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí

Architectures for soft-decision decoding of non-binary codes

En esta tesis se estudia el dise¿no de decodificadores no-binarios para la correcci'on de errores en sistemas de comunicaci'on modernos de alta velocidad. El objetivo es proponer soluciones de baja complejidad para los algoritmos de decodificaci'on basados en los c'odigos de comprobaci'on de paridad de baja densidad no-binarios (NB-LDPC) y en los c'odigos Reed-Solomon, con la finalidad de implementar arquitecturas hardware eficientes. En la primera parte de la tesis se analizan los cuellos de botella existentes en los algoritmos y en las arquitecturas de decodificadores NB-LDPC y se proponen soluciones de baja complejidad y de alta velocidad basadas en el volteo de s'¿mbolos. En primer lugar, se estudian las soluciones basadas en actualizaci'on por inundaci 'on con el objetivo de obtener la mayor velocidad posible sin tener en cuenta la ganancia de codificaci'on. Se proponen dos decodificadores diferentes basados en clipping y t'ecnicas de bloqueo, sin embargo, la frecuencia m'axima est'a limitada debido a un exceso de cableado. Por este motivo, se exploran algunos m'etodos para reducir los problemas de rutado en c'odigos NB-LDPC. Como soluci'on se propone una arquitectura basada en difusi'on parcial para algoritmos de volteo de s'¿mbolos que mitiga la congesti'on por rutado. Como las soluciones de actualizaci 'on por inundaci'on de mayor velocidad son sub-'optimas desde el punto de vista de capacidad de correci'on, decidimos dise¿nar soluciones para la actualizaci'on serie, con el objetivo de alcanzar una mayor velocidad manteniendo la ganancia de codificaci'on de los algoritmos originales de volteo de s'¿mbolo. Se presentan dos algoritmos y arquitecturas de actualizaci'on serie, reduciendo el 'area y aumentando de la velocidad m'axima alcanzable. Por 'ultimo, se generalizan los algoritmos de volteo de s'¿mbolo y se muestra como algunos casos particulares puede lograr una ganancia de codificaci'on cercana a los algoritmos Min-sum y Min-max con una menor complejidad. Tambi'en se propone una arquitectura eficiente, que muestra que el 'area se reduce a la mitad en comparaci'on con una soluci'on de mapeo directo. En la segunda parte de la tesis, se comparan algoritmos de decodificaci'on Reed- Solomon basados en decisi'on blanda, concluyendo que el algoritmo de baja complejidad Chase (LCC) es la soluci'on m'as eficiente si la alta velocidad es el objetivo principal. Sin embargo, los esquemas LCC se basan en la interpolaci'on, que introduce algunas limitaciones hardware debido a su complejidad. Con el fin de reducir la complejidad sin modificar la capacidad de correcci'on, se propone un esquema de decisi'on blanda para LCC basado en algoritmos de decisi'on dura. Por 'ultimo se dise¿na una arquitectura eficiente para este nuevo esquemaGarcía Herrero, FM. (2013). Architectures for soft-decision decoding of non-binary codes [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/33753TESISPremiad