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Satisfiability Coding Lemma

R. Paturi, P. Pudlak, F. Zane
Proceedings 38th Annual Symposium on Foundations of Computer Science  
The key idea used in these upper and lower bounds is what we call the Satis-$ability Coding Lemma. This basic lemma shows how to encode satisfying solutions of a k-CW succinctly.  ...  The first is a randomized algorithm which, with probability approaching 1, finds a satisfying assignment of a satisfiable k-CNF formula F in time O(n2 IF12"-"lk).  ...  We next prove the Satisfiability Coding Lemma and its corollaries.  ... 
doi:10.1109/sfcs.1997.646146 dblp:conf/focs/PaturiPZ97 fatcat:nzjs2sqf6ja27jlqbkwyxvyac4

On Minimum Expected Length Prefix Codes Satisfying a (d, k) Runlength-Limited Constraint

Shivkumar K Manickam, Navin Kashyap
2018 2018 International Symposium on Information Theory and Its Applications (ISITA)  
A prefix code X is said to satisfy the (d, k) runlength-limited (RLL) constraint if all the possible concatenations of the codewords of X satisfy the (d, k) RLL constraint.  ...  In this paper, the problem of constructing a minimum expected length prefix code satisfying the (d, k) RLL constraint is studied for certain (d, k) pairs.  ...  But this contradicts Lemma 7 as ϕ(X ) is a prefix code satisfying the (d, ∞) RLL constraint. So, X is RLL-maximal. Proposition 2.  ... 
doi:10.23919/isita.2018.8664350 dblp:conf/isita/ManickamK18 fatcat:k3sc3ez7tzcs3hx4qeg2plshuu

Improved Constructions of Frameproof Codes

Yeow Meng Chee, Xiande Zhang
2012 IEEE Transactions on Information Theory  
Frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data. Let be the largest cardinality of a -ary -frameproof code of length and .  ...  In this paper, we give a recursive construction for -frameproof codes of length with respect to the alphabet size .  ...  from Lemma 2.1 satisfies Property with the same of the previous code.  ... 
doi:10.1109/tit.2012.2197812 fatcat:dzsglnbnuzcnzp237rsuubfevu

Gaussian robust sequential and predictive coding

Lin Song, Jun Chen, Jia Wang, Tie Liu
2012 2012 IEEE International Symposium on Information Theory Proceedings  
We introduce two new source coding problems: robust sequential coding and robust predictive coding.  ...  For the Gauss-Markov source model with the mean squared error distortion measure, we characterize certain supporting hyperplanes of the rate region of these two coding problems.  ...  Then, for any satisfying For the purpose of proving Theorem 2, it suffices to construct satisfying the conditions in Lemma 3 such that Clearly (32) (43) Lemma 4: The minimizer to (29) is given by .  ... 
doi:10.1109/isit.2012.6283483 dblp:conf/isit/SongCWL12 fatcat:ann7f36eojdelgy4q4p4ssarey

To code, or not to code: lossy source-channel communication revisited

M. Gastpar, B. Rimoldi, M. Vetterli
2003 IEEE Transactions on Information Theory  
Lemma 4 : 4 For fixed source distribution , channel conditional distribution , and a single-letter code i) if , the second condition of Lemma 2 is satisfied if and only if the distortion measure satisfies  ...  Thus, is satisfied if and only if all three conditions in Lemma 2 are satisfied, which completes the proof.  ...  Proof of Lemma 8 Pick an arbitrary channel conditional distribution for which there exists a single-letter code that makes the overall system optimal. From Lemma 2, this implies that .  ... 
doi:10.1109/tit.2003.810631 fatcat:4ausdym2tbdbjoj7yeimgltave

On the existence of optimum cyclic burst- correcting codes

K. Abdel-Ghaffar, R. McEliece, A. Odlyzko, H. van Tilborg
1986 IEEE Transactions on Information Theory  
Abstraci-It is shown that for each integer b 2 1 infinitely many optimum cyclic b-burst-correcting codes exist, i.e., codes whose length n, redundancy r, and burst-correcting capability b, satisfy n =  ...  Some optimum codes for b = 3,4, and 5 are also studied in detail.  ...  Lemma 1: Let e(x) and p(x) satisfy conditions 1 and 2 of Theorem 1.  ... 
doi:10.1109/tit.1986.1057242 fatcat:zp347q27dvap5hkoqubsqnczdi

Universal codes of the natural numbers

Yuval Filmus, S. Barry Cooper
2013 Logical Methods in Computer Science  
A code of the natural numbers is a uniquely-decodable binary code of the natural numbers with non-decreasing codeword lengths, which satisfies Kraft's inequality tightly.  ...  As an application, we prove that the existence of a scale of codes (a well-ordered set of codes which contains a code better than any given code) is independent of ZFC.  ...  Lemma 3.2. Let c be a code and n ∈ N. There is an m = m(n) ∈ N, uniformly effective relative to c, satisfying ∞ k=m 2 −c(k) = 2 −n . Proof.  ... 
doi:10.2168/lmcs-9(3:7)2013 fatcat:mjguhq6uefborefwp5btmlnyvy

The structure of single-track Gray codes

M. Schwartz, T. Etzion
1999 IEEE Transactions on Information Theory  
All known systematic constructions for single-track Gray codes result in codes from this subclass.  ...  A subclass of single-track Gray codes, called single-track Gray codes with k-spaced heads, is also defined.  ...  The first lemma is an immediate consequence of Lemma 8. Lemma 9: and satisfy (p.1). are adjacent in . Therefore, the words and are adjacent in .  ... 
doi:10.1109/18.796379 fatcat:ud5m6vcstzf45gkxkijnkaraye

Gaussian Robust Sequential and Predictive Coding

Lin Song, Jun Chen, Jia Wang, Tie Liu
2013 IEEE Transactions on Information Theory  
We introduce two new source coding problems: robust sequential coding and robust predictive coding.  ...  Index Terms-Extremal inequality, Gauss-Markov source, minimax theorem, predictive coding, saddle point, sequential coding.  ...  Note that the optimality conditions in Lemmas 4 and 6 are clearly satisfied; therefore, to verify (38), it suf- fices to show that satisfies the optimality condition in Lemma 5.  ... 
doi:10.1109/tit.2013.2245720 fatcat:6b4u5ucwebf43lorol37b3ng2e

On optimal non-projective ternary linear codes

Mito Takenaka, Kei Okamoto, Tatsuya Maruta
2008 Discrete Mathematics  
We prove the existence of a [406, 6, 270] 3 code and the nonexistence of linear codes with parameters [458, 6, 304] 3 , [467, 6, 310] 3 , [471, 6, 313] 3 , [522, 6, 347] 3 .  ...  , 348}, n 3 (6, d) = g 3 (6, d) or g 3 (6, d) + 1 for 298 d 301 and n 3 (6, d) = g 3 (6, d) + 1 or g 3 (6, d) + 2 for 310 d 312, where n q (k, d) denotes the minimum length n for which an [n, k, d] q code  ...  Similarly, we can prove a 9 = 0 from Lemma 3.3.Lemma 3.5. (1) The spectrum of a [52, 4, 34] 3 code satisfies a i = 0 for all i / ∈ {0, 7, 8, 9, 16, 17, 18}. (2) The spectrum of a [53, 4, 35] 3 code is  ... 
doi:10.1016/j.disc.2007.07.044 fatcat:idqmdik4ejhvrdfgzw7zwhqspy

Hash Property and Fixed-rate Universal Coding Theorems [article]

Jun Muramatsu, Shigeki Miyake
2008 arXiv   pre-print
Since an ensemble of sparse matrices satisfies the hash property requirement, it is proved that we can construct universal codes by using sparse matrices.  ...  These problems are the fixed-rate lossless universal source coding problem and the fixed-rate universal channel coding problem.  ...  Fig. 1 .Fig. 2 . 12 Lossless Source Coding Mϕ -Xµ Y |X -Y -Channel Coding Lemma 1 ([ 12 , 112 Lemma 9]): For any A and u ∈ U n ,p C ({c : Au = c}) = c p C (c)χ(Au = c) = 1 |ImA|and for any u ∈ U n ,E  ... 
arXiv:0804.1183v1 fatcat:mpbr6mdklrhoriogy72f2sjvne

Identifying codes of corona product graphs [article]

Min Feng, Kaishun Wang
2013 arXiv   pre-print
If G admits an identifying code, we say that G is identifiable and denote by γ^ID(G) the minimum cardinality of an identifying code of G.  ...  In this paper, we study the identifying code of the corona product H G of graphs H and G.  ...  Then Lemma 3. 9 9 Suppose that C is an identifying code of H ⊙ G. If any identifying code of G does not satisfy (b), then |C| ≥ |V (H)| · γ ID (G) + |H(C)| + |H ′ (C)|. Proof.  ... 
arXiv:1301.4295v1 fatcat:utqlezak4fgbhndkxjilb5b35i

Deterministic Polynomial-Time Algorithms for Designing Short DNA Words [article]

Ming-Yang Kao, Henry C. M. Leung, He Sun, Yong Zhang
2012 arXiv   pre-print
., code) of n DNA strings (i.e., words) with the minimum length such that the Hamming distance between each pair of words is at least k and the n words satisfy a set of additional constraints.  ...  Previous works include those that extended results from coding theory to obtain bounds on code and word sizes for biologically motivated constraints and those that applied heuristic local searches, genetic  ...  We thank Matthew King for correcting a sign in the formula in the proof of Lemma 10.  ... 
arXiv:1201.6358v1 fatcat:kgeqku5tijgc5i6ith65lzgkqm

Deterministic polynomial-time algorithms for designing short DNA words

Ming-Yang Kao, Henry C.M. Leung, He Sun, Yong Zhang
2013 Theoretical Computer Science  
., code) of n DNA strings (i.e., words) with the minimum length such that the Hamming distance between each pair of words is at least k and the n words satisfy a set of additional constraints.  ...  Previous works include those that extended results from coding theory to obtain bounds on code and word sizes for biologically motivated constraints and those that applied heuristic local searches, genetic  ...  We thank Matthew King for correcting a sign in the formula in the proof of Lemma 10.  ... 
doi:10.1016/j.tcs.2012.12.030 fatcat:oaidgao335ht3nvgv5hx5jibau

On Open and Closed Convex Codes

Joshua Cruz, Chad Giusti, Vladimir Itskov, Bill Kronholm
2018 Discrete & Computational Geometry  
Neural codes serve as a language for neurons in the brain.  ...  Not every code is open or closed convex, however, and the combinatorial properties of a code that determine its realization by such sets are still poorly understood.  ...  We can now apply Lemma 5.7 to the "missing" codeword σ 1 , and obtain a new cover (1) that again satisfies the condition of Lemma 5.7.  ... 
doi:10.1007/s00454-018-00050-1 pmid:31571705 pmcid:PMC6768430 fatcat:6vi4eevhzffajkw4q2f3mykknq
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