A construction of F2-linear cyclic, MDS codes

In this paper we construct F2-linear codes over Fb2 with length n and dimension n−r where n=rb. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime p, let n=p−1 and utilize a partition of Zn. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to 0 as n grows (for a fixed r). When r=2 we prove that these codes are always maximum distance separable. For higher r some of them retain this property.The first author was supported by CAPES (Brazil). The work of the second author was partially supported by Spanish grants AICO/2017/128 of the Generalitat Valenciana and VIGROB-287 of the Universitat d'Alacant. The third and fourth authors were supported by NSERC (Canada). The first, third and fourth authors acknowledge support from FAPESP SPRINT grant 2016/50476-0

What do random polynomials over finite fields look like?

In this paper, we survey old and new results about random univariate polynomials over a finite field double-struck F signq. We are interested in three aspects: (1) the decomposition of a random polynomial in terms of its irreducible factors, (2) the usage of random polynomials in algorithms, and (3) the average-case analysis of algorithms that use polynomials over finite fields

Efficient pth root computations in finite fields of characteristic p

We present a method for computing pth roots using a polynomial basis over finite fields Fqm of odd characteristic p, p ≤ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension Fqm of Fq our method requires p - 1 scalar multiplications of elements in Fqm by elements in Fq . In addition, our method requires at most (p-1) [m/p] additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z 1/p , defined as the Hamming weight of z 1/p or wt (z 1/p), is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260-270, 2007) give wt (z1/3) is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find wt (z1/p = 1 always

Exact largest and smallest size of components

Golomb and Gaal [15] study the number of permutations on n objects with largest cycle length equal to k. They give explicit expressions on ranges n/(i + 1)< k ≤ n/i for i = 1,2,..., derive a general recurrence for the number of permutations of size n with largest cycle length equal to k, and provide the contribution of the ranges (n/(i + 1), n/i] for i = 1,2,..., to the expected length of the largest cycle. We view a cycle of a permutation as a component. We provide exact counts for the number of decomposable combinatorial structures with largest and smallest components of a given size. These structures include permutations, polynomials over finite fields, and graphs among many others (in both the labelled and unlabelled cases). The contribution of the ranges (n/(i + 1), n/i] for i = 1,2,..., to the expected length of the smallest and largest component is also studied

The degree of the splitting field of a random polynomial over a finite field

The asymptotics of the order of a random permutation have been widely studied. P. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group Sn is normal with mean 1 2 (log n)2 and variance 1 3 (log n)3. More recently R. Stong has shown that the mean of the order is asymptotically exp(C � n/log n + O (√n log log n/log n)) where C = 2.99047.... We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree n over a finite field

The degree of the splitting field of a random polynomial over a finite field

The asymptotics of the order of a random permutation have been widely studied. P. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group Sn is normal with mean 1 2 (log n)2 and variance 1 3 (log n)3. More recently R. Stong has shown that the mean of the order is asymptotically exp(C � n / log n + O ( √ n log log n / log n)) where C =2.99047.... We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree n over a finite field.

Handbook of finite fields

Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and each chapter is self contained and peer reviewed. The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algeb