Bikash Kumar Dey

On Network Coding for Sum-Networks

A directed acyclic network is considered where all the terminals need to recover the sum of the symbols generated at all the sources. We call such a network a sum-network. It is shown that there exists a solvably (and linear solvably) equivalent sum-network for any multiple-unicast network, and thus for any directed acyclic communication network. It is also shown that there exists a linear solvably equivalent multiple-unicast network for every sum-network. It is shown that for any set of polynomials having integer coefficients, there exists a sum-network which is scalar linear solvable over a finite field F if and only if the polynomials have a common root in F. For any finite or cofinite set of prime numbers, a network is constructed which has a vector linear solution of any length if and only if the characteristic of the alphabet field is in the given set. The insufficiency of linear net- work coding and unachievability of the network coding capacity are proved for sum-networks by using similar known results for communication networks. Under fractional vector linear network coding, a sum-network and its reverse network are shown to be equivalent. However, under nonlinear coding, it is shown that there exists a solvable sum-network whose reverse network is not solvable.

Real and "Complex" Network Codes: Promises and Challenges

As an alternative to the algebraic network codes prevalent in the literature, we consider Arithmetic Network Codes (henceforth abbreviated as ANCs), i.e., codes in which interior nodes perform finite precision arithmetic over the real or complex fields. We suggest two applications where using such codes can be advantageous. First, we demonstrate that the multi-resolution behaviour of ANCs potentially outperforms that of algebraic network codes. Second, the interfering and fading nature of wireless channels naturally results in complex linear combinations of transmissions, analogous to ANCs. We then characterize the multicast rates achievable by ANCs, and demonstrate that for high precision arithmetic these are equivalent to those obtained by algebraic network codes. We show the connection between the performance of ANCs and the numerical conditioning of network transform matrices. Using this, we obtain upper and lower bounds on the number of significant bits required to perform the finite precision arithmetic in terms of the network parameters. We compare this with simulation results for randomized and deterministic design of ANCs.

Network tomography via network coding

In this work we show how existing network coding algorithms can be used to perform network tomography, i.e., estimate network topology. We first examine a simple variant of the popular distributed random network codes proposed by (Ho et al.) and show how it can enable each network node to passively estimate the network topology upstream of it at no cost to throughput. The delays introduced by each upstream node and link can also be similarly estimated. We then consider the scenario wherein an adversary hidden in the network wishes to disrupt the estimation of network topology. We show how network error-correcting codes can be used to reliably perform network tomography if the network has sufficient connectivity, and demonstrate that network tomography is impossible otherwise.

Binary causal-adversary channels

In this work we consider the communication of information in the presence of a causal adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (x1, …, xn) bit-by-bit over a communication channel. The adversarial jammer can view the transmitted bits xi one at a time, and can change up to a p-fraction of them. However, the decisions of the jammer must be made in an online or causal manner. Namely, for each bit xi the jammers decision on whether to corrupt it or not (and on how to change it) must depend only on xj for j ≤ i. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge of x. We present a non-trivial upper bound on the amount of information that can be communicated. We show that the achievable rate can be asymptotically no greater than min{1 - H(p), (1 - 4p)+}. Here H(.) is the binary entropy function, and (1 - 4p)+ equals 1 - 4p for p ≤ 0.25, and 0 otherwise.

Some results on communicating the sum of sources over a network

We consider the problem of communicating the sum of m sources to n terminals in a directed acyclic network. Recently, it was shown that for a network of unit capacity links with either m = 2 or n = 2, the sum of the sources can be communicated to the terminals using scalar/vector linear network coding if and only if every source-terminal pair is connected in the network. We show in this paper that for any finite set of primes, there exists a network where the sum of the sources can be communicated to the terminals only over finite fields of characteristic belonging to that set. As a corollary, this gives networks where the sum can not be communicated over any finite field using vector linear network coding even though every source is connected to every terminal.

Performance Analysis of Amplify and Forward Based Cooperative Diversity in MIMO Relay Channels

Tight closed form lower bounds for the average bit error rate (BER) are derived for a dual hop cooperative network employing nonregenerative relays for the multiple input multiple output (MIMO) relay channel experiencing Rayleigh fading. The bounds are obtained for three different nonregenerative relaying schemes. The lower bounds for the BER are obtained using the moment generating function (MGF) approach by evaluating the MGF of the end-to-end equivalent signal to noise ratio (SNR) of the system. From the BER expressions obtained, we also show that the diversity order for the MIMO relay cooperative system with each relay having M antennas increases approximately by a factor M from that of a system with single-antenna relays. Simulation results confirm that the analytical expressions for the lower bounds are very tight and can thus be used to get approximate values of the BER.

Feasible alphabets for communicating the sum of sources over a network

We consider directed acyclic sum-networks with m sources and n terminals where the sources generate symbols from an arbitrary alphabet field F, and the terminals need to recover the sum of the sources over F. We show that for any co-finite set of primes, there is a sum-network which is linearly solvable only over fields of characteristics belonging to that set. We further construct a sum-network where a scalar linear solution exists over all fields other than the binary field F2. We also show that a sum-network is linearly solvable over a field if and only if its reverse network is linearly solvable over the same field.

Codes against online adversaries

In this work we consider the communication of information in the presence of an online adversarial jammer. In the setting under study, a sender wishes to communicate a message to a receiver by transmitting a codeword x = (xi,…, xn) symbol-by-symbol over a communication channel. The adversarial jammer can view the transmitted symbols xt one at a time, and can change up to a p-fraction of them. However, for each symbol xt the jammers decision on whether to corrupt it or not (and on how to change it) must depend only on Xj for j ≤ i. This is in contrast to the “classical” adversarial jammer which may base its decisions on its complete knowledge of x. More generally, for a delay parameter d e (0,1), we study the scenario in which the jammers decision on the corruption of Xi must depend solely on xj for j ≤ i − dn. In this work, the transmitted symbols are assumed to be over a sufficiently large field F. We present a tight characterization of the amount of information one can transmit in both the 0-delay and, more generally, the d-delay online setting. We show that for 0-delay adversaries, the achievable rate asymptotically equals that of the classical adversarial model. For positive values of d, we consider two types of jamming, additive and overwrite. We also extend our results to a jam-or-listen online model, where the online adversary can either jam a symbol or eavesdrop on it.

F q -linear cyclic codes over F q m : DFT approach

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