On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date

Artin-Schreier families and 2-D cycle codes

We start with the study of certain Artin-Schreier families. Using coding theory techniques, we determine a necessary and sufficient condition for such families to have a nontrivial curve with the maximum possible number of rational points over the finite field in consideration. This result produces several nice corollaries, including the existence of certain maximal curves; i.e., curves meeting the Hasse-Weil bound.We then present a way to represent two-dimensional (2-D) cyclic codes as trace codes starting from a basic zero set of its dual code. This representation enables us to relate the weight of a codeword to the number of rational points on certain Artin-Schreier curves via the additive form of Hilbert’s Theorem 90. We use our results on Artin-Schreier families to give a minimum distance bound for a large class of 2-D cyclic codes. Then, we look at some specific classes of 2-D cyclic codes that are not covered by our general result. In one case, we obtain the complete weight enumerator and show that these types of codes have two nonzero weights. In the other cases, we again give minimum distance bounds. We present examples, in some of which our estimates are fairly effcient

The Weight Enumerator of Three Families of Cyclic Codes

Cyclic codes are a subclass of linear codes and have wide applications in consumer electronics, data storage systems, and communication systems due to their efficient encoding and decoding algorithms. Cyclic codes with many zeros and their dual codes have been a subject of study for many years. However, their weight distributions are known only for a very small number of cases. In general the calculation of the weight distribution of cyclic codes is heavily based on the evaluation of some exponential sums over finite fields. Very recently, Li, Hu, Feng and Ge studied a class of $p$-ary cyclic codes of length $p^{2m}-1$, where $p$ is a prime and $m$ is odd. They determined the weight distribution of this class of cyclic codes by establishing a connection between the involved exponential sums with the spectrum of Hermitian forms graphs. In this paper, this class of $p$-ary cyclic codes is generalized and the weight distribution of the generalized cyclic codes is settled for both even $m$ and odd $m$ alone with the idea of Li, Hu, Feng, and Ge. The weight distributions of two related families of cyclic codes are also determined.Comment: 13 Pages, 3 Table