Fast Methods for Ranking Synthetic BECs
We gather existing methods that are used to compare and rank the BECs synthesized by a polar code constructor, compare them, and propose new methods that compare synthetic BECs faster.
cs.IT↗
arXiv subjects
Hsin-Po Wang
Publications and source records attributed to Hsin-Po Wang.
Optimal Self-Dual Inequalities to Order Polarized BECs
$1 - (1-x^M) ^ {2^M} > (1 - (1-x)^M) ^{2^M}$ is proved for all $x \in [0,1]$ and all $M > 1$. This confirms a conjecture about polar code, made by Wu and Siegel in 2019, that $W^{0^m 1^M}$ is more reliable than $W^{1^m 0^M}$, where $W$ is any binary erasure channel and $M = 2^m$. The proof relies on a remarkable relaxation that $m$ needs not be an integer, a cleverly crafted hexavariate ordinary differential equation, and a genius generalization of Green's theorem that concerns function composition. The resulting inequality is optimal, $M$ cannot be $2^m - 1$, witnessing how far polar code deviates from Reed--Muller code.
cs.IT↗
Density Devolution for Ordering Synthetic Channels
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 sees a substring $01$ in the binary expansion of a row index and replaces it with $10$, the user obtains a row index that is always more welcomed. We call this a "rule" and denote it by $10 \succcurlyeq 01$. In present work, we first enumerate some rules over binary erasure channels such as $1001 \succcurlyeq 0110$ and $10001 \succcurlyeq 01010$ and $10101 \succcurlyeq 01110$. We then summarize them using a "rule of rules": if $10a \succcurlyeq 01b$ is a rule, where $a$ and $b$ are arbitrary binary strings, then $100a \succcurlyeq 010b$ and $101a \succcurlyeq 011b$ are rules. This work's main contribution is using field theory, Galois theory, and numerical analysis to develop an algorithm that decides if a rule of rules is mathematically sound. We apply the algorithm to enumerate some rules of rules. Each rule of rule is capable of generating an infinite family of rules. For instance, $10c01 \succcurlyeq 01c10$ for arbitrary binary string $c$ can be generated. We found an application of $10c01 \succcurlyeq 01c10$ that is related to integer partition and the dominance order therein.
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Sub-4.7 Scaling Exponent of Polar Codes
Polar code visibly approaches channel capacity in practice and is thereby a constituent code of the 5G standard. Compared to low-density parity-check code, however, the performance of short-length polar code has rooms for improvement that could hinder its adoption by a wider class of applications. As part of the program that addresses the performance issue at short length, it is crucial to understand how fast binary memoryless symmetric channels polarize. A number, called scaling exponent, was defined to measure the speed of polarization and several estimates of the scaling exponent were given in literature. As of 2022, the tightest overestimate is 4.714 made by Mondelli, Hassani, and Urbanke in 2015. We lower the overestimate to 4.63.
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Tropical Group Testing
Polymerase chain reaction (PCR) testing is the gold standard for diagnosing COVID-19. PCR amplifies the virus DNA 40 times to produce measurements of viral loads that span seven orders of magnitude. Unfortunately, the outputs of these tests are imprecise and therefore quantitative group testing methods, which rely on precise measurements, are not applicable. Motivated by the ever-increasing demand to identify individuals infected with SARS-CoV-19, we propose a new model that leverages tropical arithmetic to characterize the PCR testing process. Our proposed framework, termed tropical group testing, overcomes existing limitations of quantitative group testing by allowing for imprecise test measurements. In many cases, some of which are highlighted in this work, tropical group testing is provably more powerful than traditional binary group testing in that it require fewer tests than classical approaches, while additionally providing a mechanism to identify the viral load of each infected individual. It is also empirically stronger than related works that have attempted to combine PCR, quantitative group testing, and compressed sensing.
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Parity-Checked Strassen Algorithm
To multiply astronomic matrices using parallel workers subject to straggling, we recommend interleaving checksums with some fast matrix multiplication algorithms. Nesting the parity-checked algorithms, we weave a product code flavor protection. Two demonstrative configurations are as follows: (A) $9$ workers multiply two $2\times 2$ matrices; each worker multiplies two linear combinations of entries therein. Then the entry products sent from any $8$ workers suffice to assemble the matrix product. (B) $754$ workers multiply two $9\times 9$ matrices. With empirical frequency $99.8\%$, $729$ workers suffice, wherein $729$ is the complexity of the schoolbook algorithm. In general, we propose probability-wisely favorable configurations whose numbers of workers are close to, if not less than, the thresholds of other codes (e.g., entangled polynomial code and PolyDot code). Our proposed scheme applies recursively, respects worker locality, incurs moderate pre- and post-processes, and extends over small finite fields.
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Complexity and Second Moment of the Mathematical Theory of Communication
The performance of an error correcting code is evaluated by its error probability, rate, and en/decoding complexity. The performance of a series of codes is evaluated by, as the block lengths approach infinity, whether their error probabilities decay to zero, whether their rates converge to capacity, and whether their growth in complexities stays under control. Over any discrete memoryless channel, I build codes such that: (1) their error probabilities and rates scale like random codes; and (2) their en/decoding complexities scale like polar codes. Quantitatively, for any constants $p,r>0$ s.t. $p+2r<1$, I construct a series of codes with block length $N$ approaching infinity, error probability $\exp(-N^p)$, rate $N^{-r}$ less than the capacity, and en/decoding complexity $O(N\log N)$ per block. Over any discrete memoryless channel, I also build codes such that: (1) they achieve capacity rapidly; and (2) their en/decoding complexities outperform all known codes over non-BEC channels. Quantitatively, for any constants $t,r>0$ s.t. $2r<1$, I construct a series of codes with block length $N$ approaching infinity, error probability $\exp(-(\log N)^t)$, rate $N^{-r}$ less than the capacity, and en/decoding complexity $O(N\log(\log N))$ per block. The two aforementioned results are built upon two pillars: a versatile framework that generates codes on the basis of channel polarization, and a calculus-probability machinery that evaluates the performances of codes. The framework that generates codes and the machinery that evaluates codes can be extended to many other scenarios in network information theory. To name a few: lossless compression, lossy compression, Slepian-Wolf, Wyner-Ziv, multiple access channel, wiretap channel, and broadcast channel. In each scenario, the adapted notions of error probability and rate approach their limits at the same paces as specified above.
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Multilinear Algebra for Minimum Storage Regenerating Codes
An $(n, k, d, α)$-MSR (minimum storage regeneration) code is a set of $n$ nodes used to store a file. For a file of total size $kα$, each node stores $α$ symbols, any $k$ nodes recover the file, and any $d$ nodes can repair any other node via each sending out $α/(d-k+1)$ symbols. In this work, we explore various ways to re-express the infamous product-matrix construction using skew-symmetric matrices, polynomials, symmetric algebras, and exterior algebras. We then introduce a multilinear algebra foundation to produce $\bigl(n, k, \frac{(k-1)t}{t-1}, \binom{k-1}{t-1}\bigr)$-MSR codes for general $t\geq2$. At the $t=2$ end, they include the product-matrix construction as a special case. At the $t=k$ end, we recover determinant codes of mode $m=k$; further restriction to $n=k+1$ makes it identical to the layered code at the MSR point. Our codes' sub-packetization level---$α$---is independent of $n$ and small. It is less than $L^{2.8(d-k+1)}$, where $L$ is Alrabiah--Guruswami's lower bound on $α$. Furthermore, it is less than other MSR codes' $α$ for a subset of practical parameters. We offer hints on how our code repairs multiple failures at once.
cs.IT↗
Multilinear Algebra for Distributed Storage
An $(n, k, d, α, β, M)$-ERRC (exact-repair regenerating code) is a collection of $n$ nodes used to store a file. For a file of total size $M$, each node stores $α$ symbols, any $k$ nodes recover the file, and any $d$ nodes repair any other node via sending out $β$ symbols. We establish a multilinear algebra foundation to assemble $(n, k, d, α, β, M)$-ERRCs for all meaningful $(n, k, d)$ tuples. Our ERRCs tie the $α/M$-versus-$β/M$ trade-off with cascade codes, the best known construction for this trade-off. We give directions on how these ERRCs repair multiple failures.
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Polar Codes' Simplicity, Random Codes' Durability
Over any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants $π,ρ>0$ such that $π+2ρ<1$, we construct a sequence of error correction codes with block length $N$ approaching infinity, block error probability $\exp(-N^π)$, code rate $N^{-ρ}$ less than the Shannon capacity, and encoding and decoding complexity $O(N\log N)$ per code block. The putative codes take uniform $ς$-ary messages for sender's choice of prime $ς$. The putative codes are optimal in the following manner: Should $π+2ρ>1$, no such codes exist for generic channels regardless of alphabet and complexity.
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Log-logarithmic Time Pruned Polar Coding
A pruned variant of polar coding is proposed for binary erasure channels. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-5}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The given per-bit complexity $\text{bC}$ is log-logarithmic in $N$, in $\text{Capacity}-R$, and in $P$; no known family of codes possesses this property. It is also the second lowest $\text{bC}$ after repeat-accumulate codes and their variants. While random codes and classical polar codes are the only two families of capacity-achieving codes whose $N$, $R$, $P$, and $\text{bC}$ were written down as explicit functions, our construction gives the third family. Then we generalize the result to: Fix a prime $q$ and fix a $q$-ary-input discrete symmetric memoryless channel. For sufficiently small $\varepsilon>0$, we construct a series of capacity achieving codes with block length $N=\varepsilon^{-O(1)}$, code rate $R=\text{Capacity}-\varepsilon$, error probability $P=\varepsilon$, and encoding and decoding time complexity $\text{bC}=O(\log\left|\log\varepsilon\right|)$ per information bit. The later construction gives the fastest family of capacity-achieving codes to date on those channels.
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Log-logarithmic Time Pruned Polar Coding on Binary Erasure Channels
A pruned variant of polar coding is reinvented for all binary erasure channels. For small $\varepsilon>0$, we construct codes with block length $\varepsilon^{-5}$, code rate $\text{Capacity}-\varepsilon$, error probability $\varepsilon$, and encoding and decoding time complexity $O(N\log|\log\varepsilon|)$ per block, equivalently $O(\log|\log\varepsilon|)$ per information bit (Propositions 5 to 8). This result also follows if one applies systematic polar coding [Arıkan 10.1109/LCOMM.2011.061611.110862] with simplified successive cancelation decoding [Alamdar-Yazdi-Kschischang 10.1109/LCOMM.2011.101811.111480], and then analyzes the performance using [Guruswami-Xia arXiv:1304.4321] or [Mondelli-Hassani-Urbanke arXiv:1501.02444].
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Polar-like Codes and Asymptotic Tradeoff among Block Length, Code Rate, and Error Probability
A general framework is proposed that includes polar codes over arbitrary channels with arbitrary kernels. The asymptotic tradeoff among block length $N$, code rate $R$, and error probability $P$ is analyzed. Given a tradeoff between $N,P$ and a tradeoff between $N,R$, we return an interpolating tradeoff among $N,R,P$ (Theorem 5). $\def\Capacity{\text{Capacity}}$Quantitatively, if $P=\exp(-N^{β^*})$ is possible for some $β^*$ and if $R=\Capacity-N^{1/μ^*}$ is possible for some $1/μ^*$, then $(P,R)=(\exp(-N^{β'}),\Capacity-N^{-1/μ'})$ is possible for some pair $(β',1/μ')$ determined by $β^*$, $1/μ^*$, and auxiliary information. In fancy words, an error exponent regime tradeoff plus a scaling exponent regime tradeoff implies a moderate deviations regime tradeoff. The current world records are: [arXiv:1304.4321][arXiv:1501.02444][arXiv:1806.02405] analyzing Arıkan's codes over BEC; [arXiv:1706.02458] analyzing Arıkan's codes over AWGN; and [arXiv:1802.02718][arXiv:1810.04298] analyzing general codes over general channels. An attempt is made to generalize all at once (Section IX). As a corollary, a grafted variant of polar coding almost catches up the code rate and error probability of random codes with complexity slightly larger than $N\log N$ over BEC. In particular, $(P,R)=(\exp(-N^{.33}),\Capacity-N^{-.33})$ is possible (Corollary 10). In fact, all points in this triangle are possible $(β',1/μ')$-pairs. $$ \require{enclose} \def\r{\phantom{\Rule{4em}{1em}{1em}}} \enclose{}\r^\llap{(0,1/2)}_\llap{(0,0)} \enclose{left,bottom,downdiagonalstrike}\r_\rlap{(1,0)} \enclose{}\r $$
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Polar Code Moderate Deviation: Recovering the Scaling Exponent
In 2008 Arikan proposed polar coding [arXiv:0807.3917] which we summarize as follows: (a) From the root channel $W$ synthesize recursively a series of channels $W_N^{(1)},\dotsc,W_N^{(N)}$. (b) Select sophisticatedly a subset $A$ of synthetic channels. (c) Transmit information using synthetic channels indexed by $A$ and freeze the remaining synthetic channels. Arikan gives each synthetic channel a score (called the Bhattacharyya parameter) that determines whether it should be selected or frozen. As $N$ grows, a majority of the scores are either very high or very low, i.e., they polarize. By characterizing how fast they polarize, Arikan showed that polar coding is able to produce a series of codes that achieve capacity on symmetric binary-input memoryless channels. In measuring how the scores polarize the relation among block length, gap to capacity, and block error probability are studied. In particular, the error exponent regime fixes the gap to capacity and varies the other two. The scaling exponent regime fixes the block error probability and varies the other two. The moderate deviation regime varies all three factors at once. The latest result [arxiv:1501.02444, Theorem 7] in the moderate deviation regime does not imply the scaling exponent regime as a special case. We give a result that does. (See Corollary 8.)
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