Repetition code

Abstract: An optimum method of coding an ensemble of messages consisting of a finite number of members is developed. A minimum-redundancy code is one constructed in such a way that the average number of coding digits per message is minimized.

Abstract: It is well known that if the data rate is chosen below the available channel capacity, error-free communication is possible. Furthermore, numerous practical error-correction coding techniques exist which can be chosen to meet the user's reliability constraints. However, a basic problem in designing a reliable digital communication system is still the choice of the actual code rate. While the popular rate-1/2 code rate is a reasonable, but not optimum, choice for additive Gaussian noise channels, its selection is far from optimum for channels where a high percentage of the transmitted bits are destroyed by interference. Code combining represents a technique of matching the code rate to the prevailing channel conditions. Information is transmitted in packet formats which are encoded with a relatively high-rate code, e.g., rate 1/2, which can be repeated to Obtain reliable communications when the redundancy in a rate-1/2 code is not sufficient to overcome the channel interference. The receiver combines noisy packets (code combining) to obtain a packet with a code rate which is low enough such that reliable communication is possible even for channels with extremely high error rates. By combining the minimum number of packets needed to overcome the channel conditions, the receiver optimizes the code rate and minimizes the delay required to decode a given packet. Thus, the receiver adapts to the actual jammer-to-signal (J/S) ratio which is critical when the level of interference J is not known a priori.

Abstract: Reliable communication protocols require that all the intended recipients of a message receive the message intact. Automatic Repeat reQuest (ARQ) techniques are used in unicast protocols, but they do not scale well to multicast protocols with large groups of receivers, since segment losses tend to become uncorrelated thus greatly reducing the effectiveness of retransmissions. In such cases, Forward Error Correction (FEC) techniques can be used, consisting in the transmission of redundant packets (based on error correcting codes) to allow the receivers to recover from independent packet losses.Despite the widespread use of error correcting codes in many fields of information processing, and a general consensus on the usefulness of FEC techniques within some of the Internet protocols, very few actual implementations exist of the latter. This probably derives from the different types of applications, and from concerns related to the complexity of implementing such codes in software. To fill this gap, in this paper we provide a very basic description of erasure codes, describe an implementation of a simple but very flexible erasure code to be used in network protocols, and discuss its performance and possible applications. Our code is based on Vandermonde matrices computed over GF(pr), can be implemented very efficiently on common microprocessors, and is suited to a number of different applications, which are briefly discussed in the paper. An implementation of the erasure code shown in this paper is available from the author, and is able to encode/decode data at speeds up to several MB/s running on a Pentium 133.

Abstract: We consider a communications scenario in which a transmitter attempts to inform a remote receiver of the state of a source by sending messages through an imperfect communications channel. There are two fundamentally different ways in which the receiver can end up being misinformed. The channel may be noisy so that symbols in the transmitted message can be received in error, or the channel may be under the control of an opponent who can either deliberately modify legitimate messages or else introduce fraudulent ones to deceive the receiver, i.e., what Wyner has called an "active wiretapper" [1]. The device by which the receiver improves his chances of detecting error (deception) is the same in either case: the deliberate introduction of redundant information into the transmitted message. The way in which this redundant information is introduced and used, though, is diametrically opposite in the two cases.For a statistically described noisy channel, coding theory is concerned with schemes (codes) that introduce redundancy in such a way that the most likely alterations to the encoded messages are in some sense close to the code they derive from. The receiver can then use a maximum likelihood detector to decide which (acceptable) message he should infer as having been transmitted from the (possibly altered) code that was received. In other words, the object in coding theory is to cluster the most likely alterations of an acceptable code as closely as possible (in an appropriate metric) to the code itself, and disjoint from the corresponding clusters about other acceptable codes.