On nuclearity of Köthe spaces

In this study we observe that the Köthe spaces Klp(A) is nuclear when it is complementedly embedded in Klq (B) for 1 p 2

Varentropy Decreases Under the Polar Transform

We consider the evolution of variance of entropy (varentropy) in the course of a polar transform operation on binary data elements (BDEs). A BDE is a pair $(X,Y)$ consisting of a binary random variable $X$ and an arbitrary side information random variable $Y$. The varentropy of $(X,Y)$ is defined as the variance of the random variable $-\log p_{X|Y}(X|Y)$. A polar transform of order two is a certain mapping that takes two independent BDEs and produces two new BDEs that are correlated with each other. It is shown that the sum of the varentropies at the output of the polar transform is less than or equal to the sum of the varentropies at the input, with equality if and only if at least one of the inputs has zero varentropy. This result is extended to polar transforms of higher orders and it is shown that the varentropy decreases to zero asymptotically when the BDEs at the input are independent and identially distributed.Comment: Presented in part at ISIT 2014. Accepted for publication in the IEEE Trans. Inform. Theory, March 201

A Packing Lemma for Polar Codes

A packing lemma is proved using a setting where the channel is a binary-input discrete memoryless channel $(\mathcal{X},w(y|x),\mathcal{Y})$, the code is selected at random subject to parity-check constraints, and the decoder is a joint typicality decoder. The ensemble is characterized by (i) a pair of fixed parameters $(H,q)$ where $H$ is a parity-check matrix and $q$ is a channel input distribution and (ii) a random parameter $S$ representing the desired parity values. For a code of length $n$, the constraint is sampled from $p_S(s) = \sum_{x^n\in {\mathcal{X}}^n} \phi(s,x^n)q^n(x^n)$ where $\phi(s,x^n)$ is the indicator function of event $\{s = x^n H^T\}$ and $q^n(x^n) = \prod_{i=1}^nq(x_i)$. Given $S=s$, the codewords are chosen conditionally independently from $p_{X^n|S}(x^n|s) \propto \phi(s,x^n) q^n(x^n)$. It is shown that the probability of error for this ensemble decreases exponentially in $n$ provided the rate $R$ is kept bounded away from $I(X;Y)-\frac{1}{n}I(S;Y^n)$ with $(X,Y)\sim q(x)w(y|x)$ and $(S,Y^n)\sim p_S(s)\sum_{x^n} p_{X^n|S}(x^n|s) \prod_{i=1}^{n} w(y_i|x_i)$. In the special case where $H$ is the parity-check matrix of a standard polar code, it is shown that the rate penalty $\frac{1}{n}I(S;Y^n)$ vanishes as $n$ increases. The paper also discusses the relation between ordinary polar codes and random codes based on polar parity-check matrices.Comment: 5 pages. To be presented at 2015 IEEE International Symposium on Information Theory, June 14-19, 2015, Hong Kong. Minor corrections to v

Channel combining and splitting for cutoff rate improvement

The cutoff rate $R_0(W)$ of a discrete memoryless channel (DMC) $W$ is often used as a figure of merit, alongside the channel capacity $C(W)$. Given a channel $W$ consisting of two possibly correlated subchannels $W_1$, $W_2$, the capacity function always satisfies $C(W_1)+C(W_2) \le C(W)$, while there are examples for which $R_0(W_1)+R_0(W_2) > R_0(W)$. This fact that cutoff rate can be ``created'' by channel splitting was noticed by Massey in his study of an optical modulation system modeled as a $M$'ary erasure channel. This paper demonstrates that similar gains in cutoff rate can be achieved for general DMC's by methods of channel combining and splitting. Relation of the proposed method to Pinsker's early work on cutoff rate improvement and to Imai-Hirakawa multi-level coding are also discussed.Comment: 5 pages, 7 figures, 2005 IEEE International Symposium on Information Theory, Adelaide, Sept. 4-9, 200

Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels

A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ is near 1 approaches $I(W)$ and the fraction for which $I(W_N^{(i)})$ is near 0 approaches $1-I(W)$. The polarized channels $\{W_N^{(i)}\}$ are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC $W$ with $I(W)>0$ and any target rate $R < I(W)$, there exists a sequence of polar codes $\{{\mathscr C}_n;n\ge 1\}$ such that ${\mathscr C}_n$ has block-length $N=2^n$, rate $\ge R$, and probability of block error under successive cancellation decoding bounded as P_{e}(N,R) \le \bigoh(N^{-\frac14}) independently of the code rate. This performance is achievable by encoders and decoders with complexity $O(N\log N)$ for each.Comment: The version which appears in the IEEE Transactions on Information Theory, July 200

Source Polarization

The notion of source polarization is introduced and investigated. This complements the earlier work on channel polarization. An application to Slepian-Wolf coding is also considered. The paper is restricted to the case of binary alphabets. Extension of results to non-binary alphabets is discussed briefly.Comment: To be presented at the IEEE 2010 International Symposium on Information Theory

Synchronization in wireless communications

The last decade has witnessed an immense increase of wireless communications services in order to keep pace with the ever increasing demand for higher data rates combined with higher mobility. To satisfy this demand for higher data rates, the throughput over the existing transmission media had to be increased. Several techniques were proposed to boost up the data rate: multicarrier systems to combat selective fading, ultra wide band (UWB) communications systems to share the spectrum with other users, MIMO transmissions to increase the capacity of wireless links, iteratively decodable codes (e.g., turbo codes and LDPC codes) to improve the quality of the link, cognitive radios, and so forth

The Causality Analysis of External Debt Service and GNP : The Case of Turkey

It is argued that debt service burden has a negative impact on investment and capital accumulation. The main reason is that the greater percentage of reserves (foreign currency) goes to meet debt service and there will be a reduction in external capital because of a decrease in creditworthiness. This paper extends the model of Cunningham (1992) and uses multivariate cointegration techniques to develop a vector error correction model useful for investigating the long-run effects of external debt service on GNP level. Moreover, the information on cointegration (Johansen ,1988 and Johansen &Juselius ,1990) in variables are taken into consideration in specifying the correct model. We apply our methodology to Turkey and show how external debt service is having a negative short -run impact on economic growth. The results also show that there is a uni-directional causal relationship between debt service and GNP level.Turkey, External Debt, Cointegration, Causality

The relationship between the F-test and the Schwarz criterion: Implications for Granger-causality tests

In applied research, the Schwarz Bayesian Information Criterion (BIC) and the F-test might yield different inferences about the causal relationships being investigated. This paper examines the relationship between the BIC and the F-tests in the context of Granger-causality tests. We calculate the F-test significance levels as a function of the model dimensionality and the sample size that would lead to the same conclusion as the BIC. We illustrate that the BIC would reject the null hypothesis of no-causality less often compared to an F-test conducted at five percent significance level for sample sizes above 50 especially when the chosen model dimensionality is small. Putting the philosophical issues aside, we suggest that the decision to choose between the F-test and the BIC should be made in view of the sample size.F-test, Schwarz Bayesian Information Criterion, Model selection, Granger-causality