Eli Haim

A note on the dispersion of network problems

Recently there has been significant interest in the analysis of finite-blocklength performance in different settings. Specifically, there is an effort to extend the performance bounds, as well as the Gaussian approximation (dispersion) beyond point-to-point settings. This proves to be a difficult task, as the performance may be governed by multiple dependent constraints. In this work we shed light on these difficulties, using the multiple-access channel as a test case. We show that a local notion of dispersion is more informative than that of dispersion regions sought after thus far. On the positive side, we show that for channels possessing certain symmetry, the dispersion problem reduces to the single-user one. Furthermore, for such channels, linear codes enable to translate single-user achievability bounds to the multiple-access channel.

Improving the MAC error exponent using distributed structure

Structured codes have been utilized in deriving the best known achievable rate regions in certain network scenarios. In this paper we demonstrate that structure can be also beneficial in terms of error exponents even in cases where there is no capacity gain. We use distributed structure, i.e., different users use codes which satisfy a nesting condition. Specifically, for the scalar Gaussian multiple-access channel we obtain an improvement over the best known achievable exponent, given by Gallager, for certain rate pairs.

Expurgation for discrete multiple-access channels via linear codes

We consider the error exponent of the memoryless multiple-access (MAC) channel. 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 we can improve upon the best known exponent for these cases as well. This approach is related to that previously proposed for the Gaussian MAC channel, and is based on “distributed structure”.

Distributed Structure: Joint Expurgation for the Multiple-Access Channel

In this paper, we obtain an improved lower bound on the error exponent of the memoryless multiple-access channel via the use of linear codes, thus demonstrating that structure can be beneficial even when capacity may be achieved via random codes. We show that if the multiple-access channel is additive over a finite field, then any error probability, and hence any error exponent, achievable by a linear code for the associated single-user channel, is also achievable for the multiple-access channel. In particular, linear codes allow to attain joint expurgation, and hence, attain the single-user expurgated exponent of the single-user channel, whenever the latter is achieved by a uniform distribution. Thus, for additive channels, at low rates, where expurgation is needed, our approach strictly improves performance over previous results, where expurgation was used for at most one of the users. Even when the multiple-access channel is not additive, it may be transformed into such a channel. While the transformation is information-lossy, we show that the distributed structure gain in some “nearly additive” cases outweighs the loss. Finally, we apply a similar approach to the Gaussian multiple-access channel. While we believe that due to the power constraints, it is impossible to attain the single-user error exponent, we do obtain an improvement over the best known achievable error exponent, given by Gallager, for certain parameters. This is accomplished using a nested lattice triplet with judiciously chosen parameters.

The importance of tie-breaking in finite-blocklength bounds

Upper bounds on the error probability in channel coding are considered, improving the RCU bound by taking into account events, where the likelihood of the correct codeword is tied with that of some competitors. This bound is compared to various previous results, both qualitatively and quantitatively; it is shown to be the tightest bound with respect to previous bounds with the same computational complexity. With respect to maximal error probability of linear codes, it is observed that when the channel is additive, the derivation of bounds, as well as the assumptions on the admissible encoder and decoder, simplify considerably.

Broadcast Channels and Input-Cost Side Information

We consider two problems: the first one is input constrained channels where the cost function depends on a state, and the state is known only to the encoder. We show that this model is similar both in theoretical aspects and in practical coding aspects to the problem of source coding with distortion side information. We explore the capacity loss due to not knowing the cost state at the decoder and show that it is small under various assumptions. This model plays an important role in the second problem we consider, which is broadcast channel. We explore the gap between the best known single letter achievable region and the true capacity region, using tools developed for the first problem.

Binary distributed hypothesis testing via Körner-Marton coding

We consider the problem of distributed binary hypothesis testing of two sequences that are generated by a doubly binary symmetric source. Each sequence is observed by a different terminal. The two hypotheses correspond to different levels of correlation between the two source components, i.e., the i.i.d. probability of the difference between the two sequences. The terminals communicate with a decision function via equal-rate noiseless links. We analyze the tradeoff between the exponential decay of the error probabilities of the hypothesis test and the communication rate. As Körner-Marton coding is known to minimize the rate in the corresponding distributed compression problem of conveying the difference sequence, it constitutes a natural candidate for the present setting. Indeed, using this scheme we derive achievable error exponents. Interestingly, these coincide with part of the optimal tradeoff without communication constraints, even when the rate is below the Körner-Marton rate for one of the hypotheses.

Tie breaking for singular channels is worth 1 Nat

We consider the finite-blocklength performance of singular channels (e.g., the binary erasure channel). At least for symmetric singular channels, it is known that the next correction term after the channel dispersion is the “constant” term. We show that for such channels, the asymptotic significance of tie breaking (i.e., making a fair decision in the case of multiple codewords of equal likelihood) is greater than for non-singular channels. Specifically, for an ensemble of codebooks, where the codewords are independent with the same marginal distribution, the constant correction term is increased by exactly one nat.

On the tightness of Marton's regions for semi-additive broadcast channels

We study cost constrained side-information channels, where the cost function depends on a state which is known only to the encoder. In the additive noise case, we bound the capacity loss due to not knowing the cost state at the decoder and show that it is small under various assumptions, and goes to zero in the limit of weak noise. This model plays an important role in the (non-degraded) broadcast channel. In the semi-additive noise case, we bound the gap between the best known single letter achievable region and the true capacity region, using tools developed for the first problem. In the limit of weak noise, we show that the bounds coincide, thus we get the complete characterization of the capacity region.