A Note on Powers in Finite Fields

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of problems from applied mathematical fields such as signal analysis, system theory, coding theory and cryptology. In this connection it is of interest to know criteria for the existence of squares and other powers in arbitrary finite fields. Making good use of polynomial division in polynomial rings over finite fields, we have examined a classical criterion of Euler for squares in odd prime fields, giving it a formulation which is apt for generalization to arbitrary finite fields and powers. Our proof uses algebra rather than classical number theory, which makes it convenient when presenting basic methods of applied algebra in the classroom.Comment: 4 page

Graphs, designs and codes related to the n-cube

For integers n 1; k 0, and k n, the graph k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2 n when n 0 .mod 4/, n 8

Graphs, designs and codes related to the n-cube

For integers n 1; k 0, and k n, the graph k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 1; 2; 3, following the results obtained for the binary codes of the n-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Qn where n is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152 159 ]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 2 n when n 0 .mod 4/, n 8

Feng-Rao decoding of primary codes

We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S. Miura, On the Feng-Rao bound for the L-construction of algebraic geometry codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P. Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition by Matsumoto and Miura requires the use of differentials which was not needed in [Andersen and Geil 2008]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the works by I.M. Duursma and R. Pellikaa

New Constant-Weight Codes from Propagation Rules

This paper proposes some simple propagation rules which give rise to new binary constant-weight codes.Comment: 4 page

The Genesis of a Theorem

We present the story of a theorem\u27s conception and birth. The tale begins with the circumstances in which the idea sprouted; then is the question\u27s origin; next comes the preliminary investigation, which led to the conjecture and the proof; finally, we state the theorem. Our discussion is accessible to anyone who knows mathematical induction. Therefore, this material can be used for instruction in a variety of courses. In particular, this story may be used in undergraduate courses as an example of how mathematicians do research. As a bonus, the proof by induction is not of the simplest kind, because it includes some preliminary work that facilitates the proof; therefore, the theorem can also serve as a nice exercise in induction. Additionally, we use well-known facts from calculus to clarify and enhance what is intrinsically a discrete problem. Making an unexpected but welcome explanatory appearance, the number e is pertinent

Remarks on Clifford codes

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes. These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which is not already a stabilizer code. We derive a necessary and sufficient condition that allows to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford codes for error groups of small order.Comment: 10 pages; submitted to Quantum Information and Computatio