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We address the problem of bounding below the probability of error under maximum likelihood decoding of a binary code with a known distance distribution used on a binary symmetric channel. An improved upper bound is given for the maximum…
This work proves new results on the ability of binary Reed-Muller codes to decode from random errors and erasures. We obtain these results by proving improved bounds on the weight distribution of Reed-Muller codes of high degrees.…
Similar to existing codes, puncturing and shortening are two general ways to obtain an arbitrary code length and code rate for polar codes. When some of the coded bits are punctured or shortened, it is equivalent to a situation in which the…
Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers,…
A polar coding scheme is introduced in this paper for the wire-tap channel. It is shown that the provided scheme achieves the entire rate-equivocation region for the case of symmetric and degraded wire-tap channel, where the weak notion of…
In this paper, we propose a novel partial order for binary discrete memoryless channels that we call the symmetric convex ordering. We show that Ar{\i}kan's polar transform preserves 'symmetric convex orders'. Furthermore, we show that…
We describe some pseudorandom properties of binary linear codes achieving capacity on the binary erasure channel under bit-MAP decoding (as shown in Kudekar et al this includes doubly transitive codes and, in particular, Reed-Muller codes).…
The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input…
Existing polarization theories have mostly been concerned with Shannon's information measures, such as Shannon entropy and mutual information, and some related measures such as the Bhattacharyya parameter. In this work, we extend…
Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both…
Polar coding is a recently proposed coding technique that can provably achieve the channel capacity. The polar code structure, which is based on the original 2x2 generator matrix, polarises the channels, i.e., a portion of the channel…
Arikan's recursive code construction is designed to polarize a collection of memoryless channels into a set of good and a set of bad channels, and it can be efficiently decoded using successive cancellation. It was recently shown that the…
Constructing a polar code is all about selecting a subset of rows from a Kronecker power of $[^1_1{}^0_1]$. It is known that, under successive cancellation decoder, some rows are Pareto-better than the other. For instance, whenever a user…
The Poltyrev bound provides a very tight upper bound on the decoding error probability when using binary linear codes for transmission over the binary symmetric channel and the additive white Gaussian noise channel, making use of the code's…
We compare the performance of short-length linear binary codes on the binary erasure channel and the binary-input Gaussian channel. We use a universal decoder that can decode any linear binary block code: Gaussian-elimination based…
We consider the problem of efficiently constructing polar codes over binary memoryless symmetric (BMS) channels. The complexity of designing polar codes via an exact evaluation of the polarized channels to find which ones are "good" appears…
We consider the $2^n$ channels synthesized by the $n$-fold application of Ar\i{}kan's polar transform to a binary erasure channel (BEC). The synthetic channels are BECs themselves, and we show that, asymptotically for almost all these…
We consider explicit polar constructions of blocklength $n\rightarrow\infty$ for the two extreme cases of code rates $R\rightarrow1$ and $R\rightarrow0.$ For code rates $R\rightarrow1,$ we design codes with complexity order of $n\log n$ in…
Achieving information-theoretic security using explicit coding scheme in which unlimited computational power for eavesdropper is assumed, is one of the main topics is security consideration. It is shown that polar codes are capacity…
We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the binary erasure channel, this coupling increases the belief propagation threshold of…