Elwyn R. Berlekamp’s research while affiliated with Berkeley Lights, Inc. and other places

... The steps are of high efficiency because we can find the root of a polynomial in polynomial time. [29] One can show that using these steps, we can solve HDLP by computing at most m DLPs, which can be done in sub-exponential time [29] with classical computer, or be done in polynomial time with quantum computer [3]. ...

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Cryptanalysis of the cryptosystems based on the generalized hidden discrete logarithm problem

... Such solution formulas have been the mathematical basis of fast algorithms to compute the roots of polynomials ( ) ∈ F [ ] of degree up to 4, where F is the finite field with elements. Algorithms using algebraic solution formulas to find the roots of polynomials of degree up till 4 are for example given in [2], [3], [6], [9], [11]. If the degree of the polynomial exceeds 4, no solution formula for the roots of ( ) in terms of radicals exists in general and other methods are used. ...

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Using Tschirnhaus transformations to find roots in quintic polynomials over F2m

... Moreover, there exist some self-dual codes with square-root-like lower bound constructed in [8, 16-18, 29, 31, 33] and the references therein. Some families of rate 1/2 ternary negacyclic codes and (1,4) [15,7,5] Optimal C (2,4) [15,5,7] Optimal C (1,6) [63, 39,9] Best-known C (2,6) [63, 36,11] Best-known constacyclic codes were developed in [28,29], which exhibit a square-root-like lower bound on their minimum distances. It is always an interesting problem that if there exists an infinite family of asymptotically good cyclic codes [26]. ...

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Constructions of binary cyclic codes with minimum weights exceeding the square-root lower bound

... Any such code is a subgroup of F~ and so partitions V~ into cosets. The classification of these cosets is an unsolved problem for most codes, the cosets of the first order Reed Muller code having received attention recently in Berlekamp (1968band 1970), 81oane and Dick (1971, Holmes (1971), and Berlekamp and Welch (1972). In this note we examine some properties of the structure code of a vector, and answer a question proposed by Sloane and Dick (1971). ...

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Coset Analysis of Reed Muller Codes Via Translates of Finite Vector Spaces

... Since char(F) ≤ q and [F : F p ] = [F : F q ] · [F q : F p ] = (q − 1) log q, we have [F : F p ] ≤ q log q. Using Berlekamp's deterministic factorization algorithm, we can find all roots of R 2 (Y 1 ) in time poly(d, q) [Ber70], [Ker09]. Each such root is retrieved as an element inF, which corresponds to a polynomial f (X) ∈ F q [X] of degree less than q − 1. ...

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Beyond the Guruswami-Sudan (and Parvaresh-Vardy) Radii: Folded Reed-Solomon, Multiplicity and Derivative Codes

... It is well-known that the minimum nonzero Hamming weight of RM(r, n) equals 2 n−r (see [22,Chapter 13], and see [8,Chapter 4] for a more direct proof), and that the nonzero minimum weight codewords in this code are the indicators of the (n − r)-dimensional affine subspaces of F n 2 . All the low Hamming weights are known in all Reed-Muller codes, and there are very few: Berlekamp and Sloane [4] (see the Addendum in this paper) and Kasami and Tokura [16] have shown that, for r ≥ 2, the only Hamming weights in RM(r, n) occurring in the range [2 n−r ; 2 n−r+1 [ are of the form 2 n−r+1 − 2 n−r+1−i , where we have i ≤ max(min(n − r, r), n−r+2 2 ). The latter has completely characterized the codewords: The corresponding functions are affinely equivalent either to x 1 · · · x r−2 (x r−1 x r + x r+1 x r+2 + · · · + x r+2l−3 x r+2l−2 ), 2 ≤ 2l ≤ n − r + 2, or to x 1 · · · x r−l (x r−l+1 · · · x r + x r+1 · · · x r+l ), 3 ≤ l ≤ min(r, n − r). ...

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Identifying codewords in general Reed-Muller codes and determining their weights

... For S = (s 1 Further, let £> n denote the dihedral group of a regular n-gon. Then we say two n-tuples S and R are related if either R -o(S) or R = a(9fll (5)) for some o e £> n -If S and R are related, then we write S ~ R. It is easily seen that ~ is an equivalence relation. ...

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Length of the 7-Number Game

... When designing decoders, taking into account only the worst case decoding complexity is suboptimal. In [2], which discusses a traditional Reed-Solomon decoder, a similar observation (experimental in that case) of error-weight dependent decoding time is made. That observation then motivates a buffered implementation of the decoder that dramatically increases its effective speed. ...

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On the average complexity of Reed-Solomon algebraic list decoders

... Such solution formulas have been the mathematical basis of fast algorithms to compute the roots of polynomials ( ) ∈ F [ ] of degree up to 4, where F is the finite field with elements. Algorithms using algebraic solution formulas to find the roots of polynomials of degree up till 4 are for example given in [2], [3], [6], [9], [11]. If the degree of the polynomial exceeds 4, no solution formula for the roots of ( ) in terms of radicals exists in general and other methods are used. ...

Reference:

Using Tschirnhaus transformations to find roots in quintic polynomials over F2m

... In 1973 Sloane [34] posed a question which remains unresolved: is there a binary self-dual doubly-even [72, 36, 16] code? The automorphism group of the extended Golay code is the 5-transitive Mathieu group M 24 of order 2 10 ·3 3 ·5·7·11·23 (see [3]), as the automorphism group of q 48 is only 2-transitive and is isomorphic to the projective special linear group PSL(2, 47) of order 2 4 · 3 · 23 · 47 [25]. The first authors to study the automorphism group of the putative [72, 26,16] code were Conway and Pless [13], in particular they focused on the possible automorphisms of odd prime order. ...

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On the Structure of the Linear Codes with a Given Automorphism