Elements Of Finite Field Research Articles

Primitive elements in finite fields and costas arrays

Primitive elements in finite fields and costas arrays

A VLSI architecture for performing finite field arithmetic with reduced table lookup

A new table lookup method for finding the log and antilog of finite field elements has been developed by Glover. The method makes use of several smaller tables, and is based on the Chinese remainder theorem. It often results in a significant reduction in the memory requirements of the problem. A VLSI architecture is developed here for a special case of this new algorithm to perform finite field arithmetic including multiplication, division, and the finding of an inverse element in the finite field.

The completion problem for partial packings

Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q2+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.

Notes on addition over a finite field using an addition table

AbstractArithmetic operations on a finite field are now often used in error control or digital signal processing and there is need for a more practical method for carrying out these operations. A detailed discussion is given of the arithmetic operations, with the elements of finite field represented as powers. In other words, the addition table which can be constructed is of the order of 1/(2 · m) of the Zech logarithmic table. A detailed discussion is given of its construction algorithm, and a practical arithmetic operation algorithm using the table is presented. The size of the addition table is estimated and a storage scheme is considered for the addition table over a finite field with index 2. It is also shown that the addition table is easily set up on a computer.

Computation of the Transition Matrix of a Linear Sequential Circuit

The general response of a linear sequential circuit involves a transition matrix in the form of a matrix A raised to a variable power. The power becomes arbitrarily high causing the direct computation of the transition matrix to become very cumbersome. This correspondence demonstrates that A, having finite-field elements, may be transformed into a matrix of real numbers. Known methods may then be applied to this real matrix to simplify computing a power of it. The result is then transformed back to the finite field which is the desired result However, numerical difficulties as mentioned in the conclusion prevent this method from being the final solution to the problem.

A theory of two-dimensional linear recurring arrays

In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximum-area matrices. We first define two-dimensional linear recurring arrays. In order to study the characteristics of two-dimensional linear recurring arrays, we also define two-dimensional linear cyclic codes. A systematic method of constructing two-dimensional linear recurring arrays having maximum-area matrices is given using the theory of two-dimensional cyclic codes. These arrays, here called \gamma \beta -arrays, may be said to be two-dimensional analogs of M -sequences. A \gamma \beta -array of area N_x \times N_y exists over GF(q) if and only if N_x N_y is equal to q^N _ 1 for some positive integer N . Many interesting characteristics of the \gamma \beta -array, such as the properties of its autocorrelation function and the properties of the characteristic arrays, are deduced and explained.

Computation with finite fields

A technique for systematically generating representations of finite fields is presented. Relations which must be physically realized in order to implement a parallel arithmetic unit to add, multiply, and divide elements of finite fields of 2 n elements are obtained. Finally, techniques for using a maximal length linear recurring sequence to modulate a radar transmitter and the means of extracting range information from the returned sequence are derived. * Operated with support from the U.S. Army, Navy and Air Force.