Combinatorics of the Two-Variable Zeta Function

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

LetXbe a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristicpbig enough. Given a local ringOp,x at a rational singular pointPofX, we attached a universal zeta function which is a rational function and admits a functional equation ifOp,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

A partial description of (complete) weight enumerators of linear self-dual codes is given. 0. Let F = Z=pZ, where p is a prime number. If C is a linear code on F of length n, i.e., a linear subspace in F n , then its (complete) weight enumerator WC is deÞned to be X u2C e Y a2F x s a(u) a e

Zeta functions enumerating normal subgroups of [formula omitted]-groups and their behavior on residue classes

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

The effectiveness of cancer treatment depends strongly on an accurate diagnosis. In this paper we propose a system for automatic and precise diagnosis of a tumor's origin based on genetic data. This system is based on a combination of coding theory techniques and machine learning algorithms. In particular, tumor classification is described as a multiclass learning setup, where gene expression values serve the system to distinguish between types of tumors. Since multiclass learning is intrinsically complex, the data is divided into several biclass problems whose results are combined with an error correcting linear block code. The robustness of the prediction is increased as errors of the base binary classifiers are corrected by the linear code. Promising results have been achieved with a best case precision of 72% when the system was tested on real data from cancer patients.

In this paper, we study a type of mixed alphabet codes, called [Formula: see text]-linear codes, where [Formula: see text] with [Formula: see text]. First, we define and discuss several weight enumerators of [Formula: see text]-linear codes, including Hamming weight, Lee weight and symmetrized weight enumerators. MacWilliams identity connects the weight enumerator of codes and their dual codes. Further, we present the MacWilliams identities about Lee weight and complete weight over the ring [Formula: see text]. Finally, several examples are presented to illustrate our main results.

This paper examines the w w -weight enumerators of weights w w with maximal symmetry over finite chain rings and matrix rings over finite fields. In many cases, including the homogeneous weight, the MacWilliams identities for w w -weight enumerators fail because there exist two linear codes with the same w w -weight enumerator whose dual codes have different w w -weight enumerators.

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

Nonlinear codes defined by quadratic forms over GF(2)

Wireless sensor network(WSN) authentication schemes are the basic necessary security tool for protecting the communication authenticity and integrity in the network. In this paper we propose general authentication schemes from linear error-correcting codes and rational functions. There are very many different random code and function based authentication schemes with the same security functionality and cost. When these different and random code and function based authentication schemes are used for various parts of one large scale wireless sensor network, the resulted WSN authentication schemes constructed in this paper can resist the compromising of large number of sensor nodes for this large scale wireless sensor network with moderate communication and storage cost. The verification per node in our novel WSN authentication schemes is very fast. Thus WSN authentication schemes based on codes and rational functions can be used to resist the pollution and DoS attacks from the adversary.

A new class of Z/sub 4/-linear codes is constructed using algebraic-geometric tools and studied. Several known Z/sub 4/-linear codes arise as special cases with an underlying rational function field. Sharp bounds for the dimension and minimum Lee weights of these codes, including what may be interpreted as the BCH and Goppa bounds for Z/sub 4/-linear codes, are derived and some efficient codes are presented.

Open problems in finite projective spaces

Bounds for generalized Hamming weights of general AG codes

Codes obtained from algebraic curves have attracted much attention from mathematicians and engineers alike since the remarkable work of Tsfasman et al. [17] who showed that the longstAnding Gilbert-Varshamov lower bound can be exceeded for alphabet sizes larger than 49. The Gilbert-Varshamov bound, established in 1952, is a lower bound on the information rate of good codes. This lower bound was not improved until 1982 with the discovery of good algebraic geometric codes. These codes are obtained from modular curves [17], but consideration of these curves is beyond the scope of this book. van Lint and Springer [21] later derived the same results as Tsfasman et al., but by using less complicated concepts from algebraic geometry. Recall that a linear code with parameters [n, k, d] q is a linear subspace of F q n of dimension k and minimum distance d.