Multiple access networks over finite fields : optimality of separation, randomness and linearity

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (leaves 93-95).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.We consider a time-slotted multiple access noisy as well as noise-free channel in which the received, transmit and noise alphabets belong to a finite field. We show that source-channel separation holds when the additive noise is independent of inputs. However, for input-dependent noise, separation may not hold. For channels over the binary field, we derive the expression for the probability of source-channel separation failing. We compute this probability to be 1/4 when the noise parameters are picked independently and uniformly. For binary channels, we derive an upper bound of 0.0776 bit for the maximum loss in sum rate due to separate source-channel coding when separation fails. We prove that the bound is very tight by showing that it is accurate to the second decimal place. We derive the capacity region and the maximum code rate for the noisy as well as noise-free channel where, code rate is defined as the ratio of the information symbols recovered at the receiver to the symbols sent by the transmitters in a slot duration. Code rate measures the overhead in transmitting in a slot under multiple access interference. We show for both noisy and noise-free channels that capacity grows logarithmically with the size of the field but the code rate is invariant with field size. For the noise-free channel, codes achieve maximum code rate if and only if they achieve capacity and add no redundancy to the shorter of the two information codewords. For the noise-free multiple access channel, we consider the cases when both transmitters always transmit in a slot, as well as when each transmitter transmits in a bursty fashion according to a Bernoulli process. For the case when both transmitters always transmit, we propose a systematic code construction and show that it achieves the maximum code rate and capacity. We also propose a systematic random code construction and show that it achieves the maximum code rate and capacity with probability tending to 1 exponentially with codeword length and field size. This is a strong coding theorem for this channel. For the case when transmitters transmit according to a Bernoulli process, we propose a coding scheme to maximize the expected code rate. We show that maximum code rate is achieved by adding redundancy at the less bursty transmitter and not adding any redundancy at the more bursty transmitter. For the noisy channel, we obtain the error exponents and hence, the expression for average probability of error when a random code is used for communicating over the channel.by Siddharth Ray.S.M

Distributed Structure: Joint Expurgation for the Multiple-Access Channel

In this work we show how an improved lower bound to the error exponent of the memoryless multiple-access (MAC) channel is attained via the use of linear codes, thus demonstrating that structure can be beneficial even in cases where there is no capacity gain. We show that if the MAC channel is modulo-additive, then any error probability, and hence any error exponent, achievable by a linear code for the corresponding single-user channel, is also achievable for the MAC channel. Specifically, for an alphabet of prime cardinality, where linear codes achieve the best known exponents in the single-user setting and the optimal exponent above the critical rate, this performance carries over to the MAC setting. At least at low rates, where expurgation is needed, our approach strictly improves performance over previous results, where expurgation was used at most for one of the users. Even when the MAC channel is not additive, it may be transformed into such a channel. While the transformation is lossy, we show that the distributed structure gain in some "nearly additive" cases outweighs the loss, and thus the error exponent can improve upon the best known error exponent for these cases as well. Finally we apply a similar approach to the Gaussian MAC channel. We obtain an improvement over the best known achievable exponent, given by Gallager, for certain rate pairs, using lattice codes which satisfy a nesting condition.Comment: Submitted to the IEEE Trans. Info. Theor

The Sender-Excited Secret Key Agreement Model: Capacity, Reliability and Secrecy Exponents

We consider the secret key generation problem when sources are randomly excited by the sender and there is a noiseless public discussion channel. Our setting is thus similar to recent works on channels with action-dependent states where the channel state may be influenced by some of the parties involved. We derive single-letter expressions for the secret key capacity through a type of source emulation analysis. We also derive lower bounds on the achievable reliability and secrecy exponents, i.e., the exponential rates of decay of the probability of decoding error and of the information leakage. These exponents allow us to determine a set of strongly-achievable secret key rates. For degraded eavesdroppers the maximum strongly-achievable rate equals the secret key capacity; our exponents can also be specialized to previously known results. In deriving our strong achievability results we introduce a coding scheme that combines wiretap coding (to excite the channel) and key extraction (to distill keys from residual randomness). The secret key capacity is naturally seen to be a combination of both source- and channel-type randomness. Through examples we illustrate a fundamental interplay between the portion of the secret key rate due to each type of randomness. We also illustrate inherent tradeoffs between the achievable reliability and secrecy exponents. Our new scheme also naturally accommodates rate limits on the public discussion. We show that under rate constraints we are able to achieve larger rates than those that can be attained through a pure source emulation strategy.Comment: 18 pages, 8 figures; Submitted to the IEEE Transactions on Information Theory; Revised in Oct 201