Network-error correcting codes using small fields

Recently, Ebrahimi and Fragouli proposed an algorithm to construct scalar network codes using small fields (and vector network codes of small lengths) satisfying multicast constraints in a given single-source, acyclic network. The contribution of this paper is two fold. Primarily, we extend the scalar network coding algorithm of Ebrahimi and Fragouli (henceforth referred to as the EF algorithm) to block network-error correction. Existing construction algorithms of block network-error correcting codes require a rather large field size, which grows with the size of the network and the number of sinks, and thereby can be prohibitive in large networks. We give an algorithm which, starting from a given network-error correcting code, can obtain another network code using a small field, with the same error correcting capability as the original code. Our secondary contribution is to improve the EF Algorithm itself. The major step in the EF algorithm is to find a least degree irreducible polynomial which is coprime to another large degree polynomial. We suggest an alternate method to compute this coprime polynomial, which is faster than the brute force method in the work of Ebrahimi and Fragouli.

A matroidal framework for network-error correcting codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. The current work attempts to establish a connection between matroid theory and network-error correcting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of network-error correcting codes to arrive at the definition of a matroidal error correcting network. An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error correcting network code if and only if it is a matroidal error correcting network associated with a representable matroid. Therefore, constructing such network-error correcting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa.

A Matroidal Framework for Network-Error Correcting Codes

Matroidal networks were introduced by Dougherty et al. and have been well studied in the recent past. It was shown that a network has a scalar linear network coding solution if and only if it is matroidal associated with a representable matroid. A particularly interesting feature of this development is the ability to construct (scalar and vector) linearly solvable networks using certain classes of matroids. Furthermore, it was shown through the connection between network coding and matroid theory that linear network coding is not always sufficient for general network coding scenarios. The current work attempts to establish a connection between matroid theory and network-error correcting and detecting codes. In a similar vein to the theory connecting matroids and network coding, we abstract the essential aspects of linear network-error detecting codes to arrive at the definition of a matroidal error detecting network (and similarly, a matroidal error correcting network abstracting from network-error correcting codes). An acyclic network (with arbitrary sink demands) is then shown to possess a scalar linear error detecting (correcting) network code if and only if it is a matroidal error detecting (correcting) network associated with a representable matroid. Therefore, constructing such network-error correcting and detecting codes implies the construction of certain representable matroids that satisfy some special conditions, and vice versa. We then present algorithms that enable the construction of matroidal error detecting and correcting networks with a specified capability of network-error correction. Using these construction algorithms, a large class of hitherto unknown scalar linearly solvable networks with multisource, multicast, and multiple-unicast network-error correcting codes is made available for theoretical use and practical implementation, with parameters, such as number of information symbols, number of sinks, number of coding nodes, error correcting capability, and so on, being arbitrary but for computing power (for the execution of the algorithms). The complexity of the construction of these networks is shown to be comparable with the complexity of existing algorithms that design multicast scalar linear network-error correcting codes. Finally, we also show that linear network coding is not sufficient for the general network-error correction (detection) problem with arbitrary demands. In particular, for the same number of network errors, we show a network for which there is a nonlinear network-error detecting code satisfying the demands at the sinks, whereas there are no linear network-error detecting codes that do the same.

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map.

Spectrum Distribution in Cognitive Radio: Error Correcting Codes Perspective

Cognitive radio is a growing zone in wireless communication which offers an opening in complete utilization of incompetently used frequency spectrum: deprived of crafting interference for the primary (authorized) user, the secondary user is indorsed to use the frequency band. Though, scheming a model with the least interference produced by the secondary user for primary user is a perplexing job. In this study we proposed a transmission model based on error correcting codes dealing with a countable number of pairs of primary and secondary users. However, we obtain an effective utilization of spectrum by the transmission of the pairs of primary and secondary users' data through the linear codes with different given lengths. Due to the techniques of error correcting codes we developed a number of schemes regarding an appropriate bandwidth distribution in cognitive radio.

Erasure error correcting codes applied to DTN communications.

The space environment has always been one of the most challenging for communications, both at physical and network layer. Concerning the latter, the most common challenges are the lack of continuous network connectivity, very long delays and relatively frequent losses. Because of these problems, the normal TCP/IP suite protocols are hardly applicable. Moreover, in space scenarios reliability is fundamental. In fact, it is usually not tolerable to lose important information or to receive it with a very large delay because of a challenging transmission channel. In terrestrial protocols, such as TCP, reliability is obtained by means of an ARQ (Automatic Retransmission reQuest) method, which, however, has not good performance when there are long delays on the transmission channel. At physical layer, Forward Error Correction Codes (FECs), based on the insertion of redundant information, are an alternative way to assure reliability. On binary channels, when single bits are flipped because of channel noise, redundancy bits can be exploited to recover the original information. In the presence of binary erasure channels, where bits are not flipped but lost, redundancy can still be used to recover the original information. FECs codes, designed for this purpose, are usually called Erasure Codes (ECs). It is worth noting that ECs, primarily studied for binary channels, can also be used at upper layers, i.e. applied on packets instead of bits, offering a very interesting alternative to the usual ARQ methods, especially in the presence of long delays. A protocol created to add reliability to DTN networks is the Licklider Transmission Protocol (LTP), created to obtain better performance on long delay links. The aim of this thesis is the application of ECs to LTP.

An upper bound for the number of certain error correcting codes /

"August 9, 1954"

Cyclic spot-error-correcting codes.

Vita.

Statistical mechanics of error-correcting codes

We investigate the performance of error-correcting codes, where the code word comprises products of K bits selected from the original message and decoding is carried out utilizing a connectivity tensor with C connections per index. Shannon's bound for the channel capacity is recovered for large K and zero temperature when the code rate K/C is finite. Close to optimal error-correcting capability is obtained for finite K and C. We examine the finite-temperature case to assess the use of simulated annealing for decoding and extend the analysis to accommodate other types of noisy channels.

Finite connectivity systems as error-correcting codes

We investigate the performance of parity check codes using the mapping onto spin glasses proposed by Sourlas. We study codes where each parity check comprises products of K bits selected from the original digital message with exactly C parity checks per message bit. We show, using the replica method, that these codes saturate Shannon's coding bound for K?8 when the code rate K/C is finite. We then examine the finite temperature case to asses the use of simulated annealing methods for decoding, study the performance of the finite K case and extend the analysis to accommodate different types of noisy channels. The analogy between statistical physics methods and decoding by belief propagation is also discussed.

Error-correcting codes that nearly saturate Shannon's bound

Gallager-type error-correcting codes that nearly saturate Shannon's bound are constructed using insight gained from mapping the problem onto that of an Ising spin system. The performance of the suggested codes is evaluated for different code rates in both finite and infinite message length.