By Venkatesan Guruswami
ISBN-10: 1601980043
ISBN-13: 9781601980045
Algorithmic leads to checklist deciphering introduces and motivates the matter of record deciphering, and discusses the crucial algorithmic result of the topic, culminating with the hot effects on reaching "list deciphering capacity." the most technical concentration is on giving an entire presentation of the hot algebraic effects reaching record deciphering capability, whereas tips or short descriptions are supplied for different works on record interpreting. Algorithmic ends up in checklist deciphering is meant for students and graduate scholars within the fields of theoretical computing device technological know-how and data thought. the writer concludes by means of posing a few attention-grabbing open questions and indicates instructions for destiny paintings.
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From the existence of Q(X, Y ) = (Y − p1 (X))(Y − p2 (X)), which ˜ Y ) be of the vanishes at all the pairs (αi , yi ), we can require that Q(X, form ˜ Q(X, Y)=Y2 +Y k j=0 q1j X j + 2k q2j X j . 1) j=0 ˜ i , yi ) = 0 pose a system of n linear equations in the The conditions Q(α variables q1j , q2j . From the existence of Q(X, Y ), we know this system ˜ has a solution. 1) by solving this linear system. The following simple lemma shows ˜ that we find. 1. If p(X) is a polynomial of degree at most k < n/6 such ˜ p(X)) ≡ 0, that p(αi ) = yi for at least n/3 values of i ∈ [n], then Q(X, ˜ or in other words Y − p(X) is a factor of Q(X, Y ).
Let f (X), g(X) be polynomials of degree at most k such that for at least t > D/r values of i ∈ [n], we have f (αi ) = yi1 and g(αi ) = yi2 . Then, Q(X, f (X), g(X)) ≡ 0. Proof. 6. If we define R(X) = Q(X, f (X), g(X)), then R(X) is a univariate polynomial of degree at most D, and for every i ∈ [n] for which f (αi ) = yi1 and g(αi ) = yi2 , (X − αi )r divides R(X). Therefore if rt > D, then R(X) has more roots (counting multiplicities) than its degree, and so it must be the zero polynomial. 2. Given an arbitrary set of n triples {(αi , yi1 , yi2 )}ni=1 from F3 and an integer parameter r 1, there exists a nonzero polynomial Q(X, Y1 , Y2 ) over F of (1, k, k)-weighted degree at most D such that Q(X, Y1 , Y2 ) has a zero of multiplicity r at (αi , yi1 , yi2 ) for all i ∈ [n], r+2 D3 provided 6k 2 > n 3 .
Even more generally, there is an abstract algebraic view of the decoding algorithm in the language of rings and ideals. The algorithms for RS codes, Chinese Remainder codes, and AG codes become just specific instantiations of this general algorithmic scheme for specific choices of the underlying ring and ideals. Details of the general algorithm for any “ideal-based” code that satisfies certain abstract axioms can be found in [28, Chapter 7] and [71]. 5 Graph-Based List-Decodable Codes In this chapter, we briefly survey some non-algebraic, graph-theoretic approaches to constructing list-decodable codes and decoding them.
ALGORITHMIC RESULTS IN LIST DECODING (Foundations and Trends(R) in Theoretical Computer Science) by Venkatesan Guruswami
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