Combinatorial lower bound for list decoding of codes on finite-field Grassmannian

2011 ◽  
Author(s):  
Rachit Agarwal
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
List Decoding

scholarly journals Improved lower bound for Diffie–Hellman problem using multiplicative group of a finite field as auxiliary group

2018 ◽  
Vol 12 (2) ◽  
pp. 101-118 ◽  
Author(s):  
Prabhat Kushwaha
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Elliptic Curves ◽  
Multiplicative Group ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Reduction Algorithm ◽  
Practical Applications ◽  
Diffie Hellman ◽  
Estimate Lower

Abstract In 2004, Muzereau, Smart and Vercauteren [A. Muzereau, N. P. Smart and F. Vercauteren, The equivalence between the DHP and DLP for elliptic curves used in practical applications, LMS J. Comput. Math. 7 2004, 50–72] showed how to use a reduction algorithm of the discrete logarithm problem to Diffie–Hellman problem in order to estimate lower bound for the Diffie–Hellman problem on elliptic curves. They presented their estimates on various elliptic curves that are used in practical applications. In this paper, we show that a much tighter lower bound for the Diffie–Hellman problem on those curves can be achieved if one uses the multiplicative group of a finite field as auxiliary group. The improved lower bound estimates of the Diffie–Hellman problem on those recommended curves are also presented. Moreover, we have also extended our idea by presenting similar estimates of DHP on some more recommended curves which were not covered before. These estimates of DHP on these curves are currently the tightest which lead us towards the equivalence of the Diffie–Hellman problem and the discrete logarithm problem on these recommended elliptic curves.

scholarly journals On NACK-Based rDWS Algorithm for Network Coded Broadcast

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 905 ◽  
Author(s):  
Sovanjyoti Giri ◽  
Rajarshi Roy
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Finite Field ◽  
Probabilistic Analysis ◽  
Queue Length ◽  
Efficiency Analysis ◽  
Entropy Analysis ◽  
Erasure Channels ◽  
Coding Strategy ◽  
High Computational Complexity

The Drop when seen (DWS) technique, an online network coding strategy is capable of making a broadcast transmission over erasure channels more robust. This throughput optimal strategy reduces the expected sender queue length. One major issue with the DWS technique is the high computational complexity. In this paper, we present a randomized version of the DWS technique (rDWS), where the unique strength of the DWS, which is the sender’s ability to drop a packet even before its decoding at receivers, is not compromised. Computational complexity of the algorithms is reduced with rDWS, but the encoding is not throughput optimal here. So, we perform a throughput efficiency analysis of it. Exact probabilistic analysis of innovativeness of a coefficient is found to be difficult. Hence, we carry out two individual analyses, maximum entropy analysis, average understanding analysis, and obtain a lower bound on the innovativeness probability of a coefficient. Based on these findings, innovativeness probability of a coded combination is analyzed. We evaluate the performance of our proposed scheme in terms of dropping and decoding statistics through simulation. Our analysis, supported by plots, reveals some interesting facts about innovativeness and shows that rDWS technique achieves near-optimal performance for a finite field of sufficient size.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

BALANCED SETS AND THE VECTOR GAME

2008 ◽  
Vol 04 (03) ◽  
pp. 339-347 ◽  
Author(s):  
ZHIVKO NEDEV ◽  
ANTHONY QUAS
Keyword(s):  
Field Theory ◽  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Log P ◽  
Prime Factorization ◽  
Minimum Cardinality ◽  
Player Game

We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x ∈ S, there is a d with 0 < d ≤ N/2 such that x ± d mod N both lie in S. We define α(N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, α(p) = Θ( log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, α(N) = min p|Nα(p).

scholarly journals A Lower Bound on List Size for List Decoding

2010 ◽  
Vol 56 (11) ◽  
pp. 5681-5688 ◽  
Author(s):  
Venkatesan Guruswami ◽  
Salil Vadhan
Keyword(s):  
Lower Bound ◽  
List Decoding ◽  
List Size

Groups generated by iterations of polynomials over finite fields

2015 ◽  
Vol 59 (1) ◽  
pp. 235-245 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Finite Fields ◽  
Mild Condition ◽  
Character Sums ◽  
Small Subgroup ◽  
Polynomials Over Finite Fields

AbstractGiven a finite field of q elements, we consider a trajectory of the map associated with a polynomial ]. Using bounds of character sums, under some mild condition on f, we show that for an appropriate constant C > 0 no N ⩾ Cq½ distinct consecutive elements of such a trajectory are contained in a small subgroup of , improving the trivial lower bound . Using a different technique, we also obtain a similar result for very small values of N. These results are multiplicative analogues of several recently obtained bounds on the length of intervals containing N distinct consecutive elements of such a trajectory.

The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x]

2019 ◽  
Vol 15 (01) ◽  
pp. 131-136 ◽  
Author(s):  
Haoli Wang ◽  
Jun Hao ◽  
Lizhen Zhang
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Prime Ideal ◽  
Polynomial Ring ◽  
Prime Power ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Multiplicative Semigroup ◽  
Associative Operation ◽  
Sharp Lower Bound

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

Large values of Dirichlet L-functions over function fields

2020 ◽  
Vol 16 (05) ◽  
pp. 1081-1109
Author(s):  
Dragan Đokić ◽  
Nikola Lelas ◽  
Ilija Vrećica
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Function Field ◽  
Function Fields ◽  
Resonance Method ◽  
Dirichlet Characters ◽  
Field Setting

In this paper, we investigate the existence of large values of [Formula: see text], where [Formula: see text] varies over non-principal characters associated to prime polynomials [Formula: see text] over finite field [Formula: see text], as [Formula: see text], and [Formula: see text]. When [Formula: see text], we provide a lower bound for the number of such characters. To do this, we adapt the resonance method to the function field setting. We also investigate this problem for [Formula: see text], where now [Formula: see text] varies over even, non-principal, Dirichlet characters associated to prime polynomials [Formula: see text] over [Formula: see text], as [Formula: see text]. In addition to resonance method, in this case, we use an adaptation of Gál-type sums estimate.

scholarly journals p-Adic estimates of abelian Artin L-functions on curves

2022 ◽  
Vol 10 ◽  
Author(s):  
Joe Kramer-Miller
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Newton Polygon ◽  
Cyclic Action ◽  
Swan Conductor ◽  
Hodge Filtration

Abstract The purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. Let X be a smooth proper curve over a finite field $\mathbb {F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$ . In this article, we prove a lower bound on the Newton polygon of the L-function $L(\rho ,s)$ . The estimate depends on monodromy invariants of $\rho $ : the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

scholarly journals Number of Directions Determined by a Set in $$\mathbb {F}_{q}^{2}$$ and Growth in $$\mathrm {Aff}(\mathbb {F}_{q})$$

2021 ◽  
Author(s):  
Daniele Dona
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Upper Bound ◽  
Finite Plane ◽  
Structural Theorem ◽  
Prime Fields ◽  
The One

AbstractWe prove that a set A of at most q non-collinear points in the finite plane $$\mathbb {F}_{q}^{2}$$ F q 2 spans more than $${|A|}/\!{\sqrt{q}}$$ | A | / q directions: this is based on a lower bound by Fancsali et al. which we prove again together with a different upper bound than the one given therein. Then, following the procedure used by Rudnev and Shkredov, we prove a new structural theorem about slowly growing sets in $$\mathrm {Aff}(\mathbb {F}_{q})$$ Aff ( F q ) for any finite field $$\mathbb {F}_{q}$$ F q , generalizing the analogous results by Helfgott, Murphy, and Rudnev and Shkredov over prime fields.

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