Finite Field Research Papers - Academia.edu (original) (raw)

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes... more

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper

Abstract. In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1 < r < q = ph, p... more

Abstract. In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1 < r < q = ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r < q and whose dual minimum distance is at least 3, has length at least (r − 1)q + (p − 1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one. 1.

The recently introduced Galois/Counter Mode (GCM) of op- eration for block ciphers provides both encryption and message authenti- cation, using universal hashing based on multiplication in a binary finite field. We analyze its security... more

The recently introduced Galois/Counter Mode (GCM) of op- eration for block ciphers provides both encryption and message authenti- cation, using universal hashing based on multiplication in a binary finite field. We analyze its security and performance, and show that it is the most ecient mode of operation for high speed packet networks, by using a realistic model of a network

I. INTRODUCTION Discrete transforms play a very important role in Engineer-ing. Particularly significant examples are the well known Dis-crete Fourier Transform (DFT) and the Z Transform [1], which have found many applications in several... more

I. INTRODUCTION Discrete transforms play a very important role in Engineer-ing. Particularly significant examples are the well known Dis-crete Fourier Transform (DFT) and the Z Transform [1], which have found many applications in several areas, specially in the field of ...

This paper examines finite field trigonometry as a tool to construct digital trigonometric transforms. In particular, by using properties of k-cosine function over a Galois field, the finite field discrete cosine transform is introduced.... more

This paper examines finite field trigonometry as a tool to construct digital trigonometric transforms. In particular, by using properties of k-cosine function over a Galois field, the finite field discrete cosine transform is introduced. The finite field DCT pair in GF(p) is defined, having blocklengths that are divisors of (p+1)/2. A special case is the Mersenne finite field DCT, defined when p is Mersenne prime. In this instance block lengths that are power of two are possible and radix-two fast algorithms can be used to compute the transform.

The design and implementation of lossless audio signal processing using Finite Field Transforms is discussed. Finite field signal processing techniques are described. The effects of filter length and coefficient accuracy are also... more

The design and implementation of lossless audio signal processing using Finite Field Transforms is discussed. Finite field signal processing techniques are described. The effects of filter length and coefficient accuracy are also discussed. Finite field transform algorithms which would be suitable for lossless signal processing are presented

Let F q [ε] := F q [X]/(X 4 − X 3) be a finite quotient ring where ε 4 = ε 3 , with F q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve... more

Let F q [ε] := F q [X]/(X 4 − X 3) be a finite quotient ring where ε 4 = ε 3 , with F q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over F q [ε], ε 4 = ε 3 of characteristic p = 2, 3 given by homogeneous Weierstrass equation of the form Y 2 Z = X 3 + aXZ 2 + bZ 3 where a and b are parameters taken in F q [ε]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve E a,b (F q [ε]) and we will show that E π0(a),π0(b) (F q) and E π1(a),π1(b) (F q) are two elliptic curves over the finite field F q , such that π 0 is a canonical projection and π 1 is a sum projection of coordinate of element in F q [ε]. Precisely, we give a classification of elements in elliptic curve over the finite ring F q [ε].

In this paper, we construct four binary linear codes closely connected with certain exponential sums over the finite field F_q and F_q-{0,1}. Here q is a power of two. Then we obtain four recursive formulas for the power moments of... more

In this paper, we construct four binary linear codes closely connected with certain exponential sums over the finite field F_q and F_q-{0,1}. Here q is a power of two. Then we obtain four recursive formulas for the power moments of Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of the exponential sums obtained earlier.

In this paper, we introduced new methods in implementing ultra-fast-efficient BCH decoder that frequently used in many applications. A Reformulated inversionless-Berlekamp-Massey algorithm is adopted in order to eliminate the finite-field... more

In this paper, we introduced new methods in implementing ultra-fast-efficient BCH decoder that frequently used in many applications. A Reformulated inversionless-Berlekamp-Massey algorithm is adopted in order to eliminate the finite-field inverter and to reduce the hardware complexity. Furthermore, we proposed a Direct reformulated-inversionless Berlekamp-Massey algorithm (DriBM). While in the Chien Search stage, the Constant-Factor Multiplication-Free Matrix transform is also introduced

As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion modules, i.e. non-trivial torsion modules whose global... more

As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra with coefficients in the finite field of p elements is a unique factorization domain in the sense of Chatters.