>parameter 𝑒, find all univariate polynomials 𝑝 of degree at most k such that y i = 𝑝(xi) for all but.at most 𝑒 values of i∈{1,. ,𝑛}. We give an algorithm that solves this problem for 𝑒<𝑛‒(𝑘𝑛)^{1/2}, which improves. the previous best result [#SudanDecoding], .>

#SudanDecoding-3. "1. Introduction. 1.1. Problem Statetement. Problem 1. INPUT: A field 𝐹; 𝑛 distinct pairs of elements (x_𝑖,𝑦_𝑖): 𝑖∈{1,. ,𝑛}} from 𝐹×𝐹; and integers 𝑑 and 𝑡. OUTPUT: A list of all functions 𝑓: 𝐹⟶𝐹 satisfying

#SudanDecoding-7."Ar et al. [Proceedings of the 33rd Annual IEEE Symposium on Foundations of #ComputerScience 1992,” pp. 503–512] do provide some solutions to the reconstruction problem, but not in its full generality. In particular, they restrict the nature of the input word. >

#SudanDecoding-8. "2. Algorithm.We now present our algorithm for solving the problem given in Section 1.1. Definition 1 (Weighted Degree). For weights 𝑤_𝑥, 𝑤_𝑦∈𝓩⁺, the (𝑤_𝑥,𝑤_𝑦)-weighted degree of a monomial 𝑄(𝑥,𝑦)=Σ_{𝑖𝑗}𝑞_{𝑖𝑗}𝑥^𝑖𝑦^𝑗 is the maximum, over.>

#SudanDecoding-9. "3. Analysis.In order to prove that the algorithm above can be run in polynomial time and works correctly, we need to prove the following set of claims. In all the following claims, we fix the set of pairs {(𝑥_1,𝑦_1),. ,(𝑥_𝑛,𝑦_𝑛)}. >

#NiebuhrCodeCryptography-1.[#AppliedMath .#Cryptography.#OverbeckSendrier.#SudanDecoding]. Attacking and Defending #CodeBasedCryptosystems: .Towards secure and efficient cryptographic applications.based on #ErrorCorrectingCodes. Robert Niebuhr. Dissertation.TU #Darmstadt, 2012

#SudanDecoding-1.[#AppliedMath.#DvirKoppartySarafSudan.#SauermannWigderson]. Journal of Complexity 13, 180–193 (1997). Decoding of #ReedSolomonCodes beyond the #ErrorCorrection[-]Bound. Madhu Sudan (#IBM). M.t. #Reed-#Solomon-Dekodierung über die Fehlerkorrekturschranke hinaus

#GuruswamiSudanDecoding-1.[#AppliedMath.#DvirKoppartySarafSudan.#SudanDecoding]. IEEE 39th Annual Symposium on Foundations of #ComputerScience - Palo Alto, #USA (8-11 Nov. 1998)]. Improved Decoding of Reed-Solomon and Algebraic-Geometric Codes. Venkatesan Guruswami .Madhu Sudan.>

#KushilevitzMansour-1.[#AppliedMath.#FriedgutKalaiNaor.#RouzeWirthHaonanZhang.#SudanDecoding]. SIAM Journal on #Computing 22(6), 1331‒1348 (1993). Learning decision trees using the Fourier spectrum. Eyal Kushilevitz (#Technion/Haifa/#Israel).Yishay Mansour (#Harvard University)

#ArLiptonRubinfeldSudan-1.[#AppliedMath.#RubinfeldSudan.#SauermannWigderson.#SudanDecoding]. SIAM Journal on #Computing 28(2), 487-510 (1998). Reconstructing algebraic functions from mixed data. Sigal Ar.Richard J. Lipton.(#Princeton/#USA).Ronitt Rubinfeld (#Cornell University).>

#OverbeckSendrier-1.[#AppliedMath.#GoldreichSudan2006.#SudanDecoding.#Survey]. pp. 95-145 in.D.J. Bernstein, J. Buchmann, E. Dahmen (eds.) Post-Quantum #Cryptography(2008). #CodeBasedCryptography. Raphael Overbeck(EPFL/#Switzerland). icolas Sendrier(#INRIA Rocquencourt/#France)>

#SudanDecoding-2. "Received [1996-08-31]. We present a randomized #algorithm which takes as input 𝑛 distinct points {(x_𝑖,y_𝑖): 𝑖∈{1,. ,𝑛}} from 𝐹×𝐹 (where 𝐹 is a field) and integer parameters 𝑡 and 𝑑 and returns a list of all univariate polynomials 𝑓 over 𝐹 . >.

#SudanDecoding-5."It is reasonable to ask for a solution to the reconstruction problem which runs in polynomial time, when the output size is bounded. Here we solve the τ-reconstruction problem for #ReedSolomonCodes for exactly the same values of the parameters δ and τ for which>.

#SudanDecoding-6. "1.3 Previous Work.As mentioned in the previous section, the τ-#error-correcting problem is well-studied and we will not describe past work in that problem here. The definition of the τ-reconstruction problem used here is based on definitions of .>.

#OrganickDumasAng-11.[#SudanDecoding]."Next, we created a coding scheme to convert digital information to DNA sequences and back to digital information. Similar to prior work, our approach employs concatenated codes with #ReedSolomon(RS) as the outer code (Fig. 2b). (However,.>.

#SudanDecoding-4. [I omit the rehearsal of basic facts from #CodingTheory.]. The case of recovering from an #error larger than the #ErrorCorrection[-]capacity of a code has not attracted the same amount of attention and significantly less is known about this problem. >.