#Eldar q emulator plus#
In this paper we show that for ℓ < k satisfying ( k – ℓ) ∤ k, ( p, μ)-denseness plus a minimum ( ℓ + 1)-vertex-degree αn k– ℓ–1 guarantees Hamilton ℓ-cycles, but requiring a minimum ℓ-vertex-degree Ω( n k– ℓ) instead is not sufficient. This is believed to be the weakest form of quasi-randomness in k-graphs and also known as linear quasi-randomness. In the trace reconstruction problem, an unknown source string x ∊ n given access to traces of x.Ī k-graph H is called ( p, μ)-dense if for all not necessarily disjoint sets A 1, …, A k ⊆ V( H) we have e( A 1, …, A k) ≥ p| A 1| ⃛ | A k| – μ| V( H)| k. This is in stark contrast to quadratic lower bounds for the worst case.
In particular, it covers arbitrary p-norms, higher dimensions, and weights affecting the area of influence of each point multiplicatively. As an additional technical contribution, we show an upper bound on the number of non-empty Voronoi regions, that holds for points with random positions in a very general setting. On the other hand, modeling locality with an underlying geometry leads to small unsatisfiable subformulas, which can be found within polynomial time.Ī key ingredient for the result on geometric random k-SAT can be found in the complexity of higher-order Voronoi diagrams. This implies intractability of these instances for modern SAT-solvers. Our findings show that heterogeneity alone does not make SAT easy as heterogeneous random k-SAT instances have superpolynomial resolution size. To understand the impact of these two properties on SAT, we study the proof complexity of random k-SAT models that allow to control heterogeneity and locality. Two characteristic properties seem to be prevalent in the majority of real-world SAT instances, heterogeneous degree distribution and locality. This disparity between theory and practice is believed to be a result of inherent properties of industrial SAT instances that make them tractable. Satisfiability is considered the canonical NP-complete problem and is used as a starting point for hardness reductions in theory, while in practice heuristic SAT solving algorithms can solve large-scale industrial SAT instances very efficiently. This gives a more flexible choice of inner codes and hence we are able to improve the lower bound on R q.
#Eldar q emulator code#
In our concatenation, the inner code is not necessarily a perfect q-hash code. Our idea is based on a modified concatenation differing from the concatenation where both the inner and outer codes are separated. Although we are not able to prove that our construction can beat the probabilistic method for all q with q (mod 4) ≠ 2, the fact that our construction beat the probabilistic method for both small and large q sheds light on that our new construction might beat the previous lower bound for all q with q (mod 4) ≠ 2. In this paper we show that this probabilistic lower bound can be improved for q = 4, 8 and all odd integers between 5 and 25, 1 and all sufficiently large q with q (mod 4) ≠ 2. This improved lower bound on R 3 was discovered in 1988 and there has been no progress of this lower bound on R q for more than 30 years despite of some work on upper bounds on R q. This is still the best-known lower bound so far except for the case q = 3 for which Körner and Matron found that the concatenation technique could lead to perfect 3-hash codes that could beat this probabilistic lower bound. Lastly we show that the ½-rate limitation does not hold for affine codes by giving an explicit affine code of rate 1 – ∊ which can efficiently correct a constant fraction of insdel errors. We complement our existential results with an efficient synchronization-string-based transformation that converts any asymptotically-good linear code for Hamming errors into an asymptotically-good linear code for insdel errors. (edit) distance trade-off of linear insdel codes. We prove novel outer bounds and existential inner bounds for the rate vs.
This identifies rate 1/2 as a sharp threshold for recovery from deletions for linear codes, and reopens the quest for a better understanding of the capabilities of linear codes for correcting insertions/deletions. We disprove this and show the existence of binary linear codes of length n and rate just below 1/2 capable of correcting Ω( n) insertions and deletions. Previously it was (erroneously) reported that more generally no non-trivial linear codes correcting k deletions exist, i.e., that the ( k + 1)-fold repetition codes and its rate of 1/( k + 1) are basically optimal for any k. Linear codes that can correct even a single deletion are limited to have information rate at most 1/2 (achieved by the trivial 2-fold repetition code). We call such codes linear/affine insdel codes. This paper studies linear and affine error-correcting codes for correcting synchronization errors such as insertions and deletions.
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