Combinatorial Structure of Finite Fields with Two Dimensional Modulo Metrics? an-Sim´ on2 , Miguel A. Borges-Trenard3, Edgar Mart´ınez-Moro1, F. Javier Gal´ and Mijail Borges-Quintana3 1
Dpto. Matem´ atica Aplicada Fundamental, Universidad de Valladolid. Valladolid, 47002 Spain
[email protected] 2 Dpto. Organizaci´ on y Gesti´ on de Empresas, Universidad de Valladolid. Valladolid, 47002 Spain
[email protected] 3 Departmento de Matem´ aticas. Facultad de Ciencias, Universidad da Oriente. Santiago de Cuba, 90500 Cuba {mborges,mijail}@csd.uo.edu.cu
Abstract. This paper shows the connection between the combinatorial structure of two dimensional metrics over finite fields ( Shortly, Mannheim and Hexagonal metrics) and some group actions defined over them. We follow the well known approach of P. Delsarte [9] to this problem through the construction of association schemes. Association schemes based on this distances are the basic tools we propose to deal with the metric properties of codes defined over two dimensional metrics and their parameters. We note that some examples of cyclotomic association schemes (which we call M schemes and H schemes respectively) fit properly as weakly metric schemes for these metrics.
1
Introduction
In this section we review briefly some basic facts about Gaussian integers and Einsestein-Jacobi integers as well as two dimensional modulo metrics defined over finite fields, for an extensive account see [7,8]. The interest on complex integers in coding theory arises from the fact that they allow us to code QAM(Quadratic Amplitude Modulation) signal spaces. Some of the constructions shown in this paper can be seen also in [14] for the Mannheim metric. 1.1
Gaussian and Einsestein-Jacobi Numbers
In this paper we consider a simple generalization of an integer. We say an algebraic number is an algebraic integer if it is a root of a monic polynomial whose ?
First and second authors are supported by Junta de Castilla y Le´ on project “Construcciones criptogr´ aficas basadas en c´ odigos correctores”, first one is also supported by Dgicyt PB97-0471.
M. Walker (Ed.): IMA - Crypto & Coding’99, LNCS 1746, pp. 45–55, 1999. c Springer-Verlag Berlin Heidelberg 1999
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Edgar Mart´ınez-Moro et al.
coefficients are rational integers. Most common examples of algebraic integers are the roots of the equations: i2 + 1 = 0, ρ2 + ρ + 1 = 0 The set of Gaussian integers is a subset of complex numbers whose real and imaginary part are integers, ie. ZZ[i], indeed they are quadratic integers since α = a + bi ∈ ZZ[i] is a root of φ2 − 2aφ − a2 − b2 = 0. The set of Einsestein-Jacobi integers are just the subset of the complex numbers given √ by ZZ[(−1+i 3)/2] = ZZ[ρ]. Again they are quadratic integers since α = a+bi ∈ ZZ[ρ] is the root of the equation:φ2 − (2a − b)φ + a2 − ab + b2 = 0. ¯ . The square norm of and For α ∈ C[i], we will denote its conjugate by α element is just kαk = αα. ¯ A basic fact in arithmetic is that the units on ZZ[i] are {±1, ±i} and the units on ZZ[ρ] are {±1, ±ρ, ±(1 + ρ)} and both are unique factorization domains. We say that two numbers are associated if they differ in the product by one unit. Any number whose norm is a prime is also a prime (the converse is false). For a reference to these facts see [6]. 1.2
Embedding Finite Fields in 2-dim Metrics
Case A: Gaussian Integers Let π be an element whose norm is a prime integer p, and p = 2 or p ≡ 1 mod 4. It is well known (Fermat’s two square theorem) that p can be written as: π where π = a + ib (not unique). p = a2 + b2 = π¯
(1)
If we denote by ZZ[i]π the set of Gaussian integers modulo π, we define the modulo function ν : ZZ[i] → ZZ[i]π associating to each class in ZZ[i]π its representant with smallest norm : ν(ξ) = r where ξ = qπ + r and krk = min{kβk | β = ξ
mod π}
(2)
This can be done because ZZ[i] is and Euclidean domain. The quotient q can be π calculated as [ α¯ p ] where [x] denotes the Gaussian integer with closest real and imaginary part to x. Taking the carrier set of GF (p) as {0, 1, . . . , p − 1} ⊂ ZZ, we can restrict to GF (p) the application ν so that it induces an isomorphism [7] ν : GF (p) → ZZ[i]π given by: g¯ π π (3) For g ∈ GF (p) ν(g) = g − p Therefore GF (p) and ZZ[i]π are mathematically equivalent but ZZ[i]π offers technical advantages for coding two-dimensional signal constellations [7]. Remark 1 In the case p ≡ 3 mod 4 π = p and the isomorphism above does not apport any relevant information over GF (p). For this type of primes −1 is a quadratic non residue of p, hence we get the following isomorphism between GF (p2 ) and ZZ p [i] where : n −(p − 1) (p − 1) o , . . . , −1, 0, 1, . . . , (4) ZZ p [i] = k + il | k, l ∈ 2 2 constructing GF (p2 ) with the irreducible polynomial x2 + 1.
Combinatorial Structure of Finite Fields
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Case B: Einsestein-Jacobi Integers In this case it is enough note that the equation (2) is valid if we denote by [·] the operation rounding to the closest Eisenstein-Jacobi integer. Clearly the size of the field must be either GF (p) for p ≡ 1 mod 6 or GF (p2 ) for p ≡ 5 mod 6 1.3
Metrics over ZZ[δ]]/πZZ[δ]
From now on δ will denote either i or ρ and π a prime in the respective set of integers.(Note that π must be a square of a prime in the exceptional cases given by p ≡ 3 mod 4 in Mannheim case and p ≡ 5 mod 6 in hexagonal case.) Mannheim Metric Let α, β ∈ ZZ[i]π and let γ = α−β mod π. The Mannheim weight of γ is defined as: ωM (γ) = |
