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Codes over Rings of Size Four, Hermitian Lattices, and Corresponding Theta Functions
Abstract
Let K = Q(root-l) be an imaginary quadratic field with ring of integers O-K, where l is a square free integer such that l = 3 mod 4, and let C = [n, k] is a linear code defined over O-K/2O(K). The level l theta function Theta(Lambda l)(C) of C is defined on the lattice Lambda(l)(C) := {x is an element of O-K(n) : rho(l)(x) is an element of C}, where rho(l) : O-K -> O-K/2O(K) is the natural projection. In this paper, we prove that: i) for any l, l ' such that l <= l ', Theta(Lambda l) (q) and Theta(Lambda l ') (q) have the same coefficients up to q (l+1/4), ii) for l >= 2(n+1)(n+2)/n - 1, Theta(Lambda l)(C) determines the code C uniquely, iii) for l < 2(n+1)(n+2)/n - 1, there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to Theta(Lambda l)(C).
... As shown in [3], for a given choice of quadratic field, two codes with different symmetric weight enumerators can generate the same theta series. Building on from the research on codes and lattices over imaginary quadratic fields in [9], we generalise similar results for real quadratic fields. This allows us to find conditions on the size of d relative to the length of the codes, which are sufficient for non-equivalent codes to generate a unique theta series. ...
... If d > 0 the field K is a real quadratic field as explored in [1], and if d < 0 the field K is an imaginary quadratic field as seen in [3,9]. ...
... For a proof that this is a lattice, refer to Lemma 3.2 in [1]. Note, as in [1,3,9], we have omitted the scaling by √ 2 typically seen when using construction A to ensure the lattice is integral. ...
Using codes defined over F 4 and F 2 × F 2 , we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields Q ( d ) with d ≡ 1 mod 4 a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of d . For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.
... For examples of non-equivalent codes corresponding to the same weight enumerator the reader can check [13], [14], and [12]. ...
... and substitute them in Eq. (14). Comparing the coefficients of monomials on both sides of the equation shows that Eq. (14) is equivalent to the system of n − d + 1 linear equations in the n − d + 1 variables a 0 , a 1 , . . . ...
... An example is when the necessary condition n ≤ q + k + 1 is not satisfied. Equation (14) leads to another definition of the zeta polynomial for a code C. Let a 0 , a 1 , . . . , a n−d be the unique constants such that ...
This is a survey on weight enumerators, zeta functions and Riemann hypothesis for linear codes.
... We aim to find MacWilliams- like identities in such cases and explore to what extent the theta functions of these lattices determine the codes. The last question was studied in [2] and [10] for p = 2. This paper is organized as follows. ...
... For general odd p, among the p 2 lattices, there are (p+1) 2 4 associated theta series. In section 4, we address a special case of a general problem of the construction of lattices: the injectivity of Construction A. For codes defined over an alphabet of size four (regarded as a quotient of the ring of integers of an imaginary quadratic field), the problem is solved completely in [10]. The analogous questions are asked for codes defined over F p 2 or F p × F p . ...
... In contrary to results in [10] we did not attempt to find explicit bounds for . However, for a given small p it is possible such bounds can be determined using similar techniques as in [10]. ...
... The first question is answered for p = 2 in [7] and for general p in [9]. For a code C defined over R and for all admissible , such that > , we have that ...
... Suppose p = 2 and let C be a code of length n defined over R = F 2 × F 2 or F 4 . In [7], which is included here as Theorem 3, for large enough, non-equivalent codes have different theta functions. Interestingly, the method used in the proof of that theorem does not work for larger primes, though in [9] we were able to generalize the above method to begin to describe the situations (in terms of and n) where more examples of different codes with the same theta function could exist. ...
... In this section we give a brief overview of codes over imaginary quadratic fields. Most of the material of this section can be found in [7] or [9]. ...
In this paper we continue the study of codes over imaginary quadratic fields and their weight enumerators and theta functions. We present new examples of non-equivalent codes over rings of characteristic p=2 and p=5 which have the same theta functions. We also look at a generalization of codes over imaginary quadratic fields, providing examples of non-equivalent pairs with the same theta function for p=3 and p=5.
... where α is a primitive element of F 4 . The permutation automorphism group is isomorphic to the group with GAP identity [24,12]. In this case PAut (C) → Aut (X ). ...
... i): p = 0 : (2, 1), (4, 2), (3, 1), (4, 1), (8, 2), (8, 3), (7, 1), (21, 1), (14,2), (6,2), (12,2), (9, 1), (8, 1), (8,5), (16,11), (16,10), (32, 9), (30, 2), (42, 3), (12,4), (16,7), (24, 5), (18, 3), (16,8), (48, 33), (48, 48). ii): p = 3 : (2, 1), (4, 2), (3, 1), (4, 1), (8, 2), (8, 3), (7, 1), (14,2), (6, 2), (8, 1), (8,5), (16,11), (16,10), (32, 9), (30, 2), (16,7), (16,8), (6,2). . ...
... i): p = 0 : (2, 1), (4, 2), (3, 1), (4, 1), (8, 2), (8, 3), (7, 1), (21, 1), (14,2), (6,2), (12,2), (9, 1), (8, 1), (8,5), (16,11), (16,10), (32, 9), (30, 2), (42, 3), (12,4), (16,7), (24, 5), (18, 3), (16,8), (48, 33), (48, 48). ii): p = 3 : (2, 1), (4, 2), (3, 1), (4, 1), (8, 2), (8, 3), (7, 1), (14,2), (6, 2), (8, 1), (8,5), (16,11), (16,10), (32, 9), (30, 2), (16,7), (16,8), (6,2). . ...
Let \X be an algebraic curve of genus defined over a field
\F_q of characteristic . From \X, under certain conditions, we can
construct an algebraic geometry code C. If the code C is self-orthogonal
under the symplectic product then we can construct a quantum code Q, called a
QAG-code. In this paper we study the construction of such codes from curves
with automorphisms and the relation between the automorphism group of the curve
\X and the codes C and Q.
... In section 4, we address a special case of a general problem of the construction of lattices: the injectivity of Construction A. For codes defined over an alphabet of size four (regarded as a quotient of the ring of integers of an imaginary quadratic field), the problem is solved completely in [7]. We expect that similar results as for p = 2 hold also for odd primes. ...
... to get θ Λ (C) (q) = swe C (θ Λ 0,0 (q), θ Λ 1,0 (q), θ Λ 0,1 (q)). These three theta functions are referred to as A d (q), C d (q), and G d (q) in [2] and [7]. ...
... In [7] we determine explicit bounds for the above theorems for prime p = 2. ...
Let be a square-free integer congruent to 3 mod 4 and the ring of integers of the imaginary quadratic field K=Q(\sqrt{-\ekk})> Codes C over rings determine lattices over K. If then the ring is isomorphic to or . Given a code C over , theta functions on the corresponding lattices are defined. These theta series can be written in terms of the complete weight enumerators of C. We show that for any two the first terms of their corresponding theta functions are the same. Moreover, we conjecture that for there is a unique symmetric weight enumerator corresponding to a given theta function. We verify the conjecture for primes , and small n.
... For examples of non-equivalent codes corresponding to the same weight enumerator the reader can check [12], [13], and [14]. ...
... We write the equations Eq. (14) in the form ...
This is a survey on weight enumerators, zeta functions and Riemann hypothesis for linear and algebraic-geometry codes.
... For examples of non-equivalent codes corresponding to the same weight enumerator the reader can check [12], [13], and [14]. ...
... We write the equations Eq. (14) in the form ...
... Γ is of maximal rank r = 2g or equivalently V 2g R /Γ being compact) is well studied in the literature [17,26,3] and the corresponding space F 2,ν,H Γ 2g ,χ (V) has a high interest not only on its own, but also in the light of the remarkable implications for both pure mathematics and mathematical physics. It is closely connected to number theory and abelian varieties [25,23,5,18], representation theory [10,22], spectral analysis [15,11,12], cryptography and coding theory [21,24], chaoticity of a shift operator [13,14] and quantum field theory [6,8]. Under the cocycle (Riemann-Dirac quantization (RDQ)) condition ...
We are interested in the -holomorphic automorphic functions on a g-dimensional complex space endowed with a positive definite hermitian form and associated to isotropic discrete subgroups of rank . We show that they form an infinite reproducing kernel Hilbert space which looks like a tensor product of a theta Fock-Bargmann space on and the classical Fock-Bargmann space on . Moreover, we provide an explicit orthonormal basis using Fourier series and we give the expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics.
... Holomorphic theta functions, associated to given lattice Γ (of maximal rank) in the d-dimensional complex space C d and given mapping χ on Γ with values in unit circle of C, is well studied in the literature [1,5,6,7,9,14,15,17,21,24]. They play important roles in many fields of mathematics, technology and especially in number theory and abelian varieties [13,22,5,18], cryptography and coding theory [20,23], as well as in quantum field theory [6]. Their existence provided a Riemann-Dirac quantization condition and constitute a finite dimensional vector space, whose dimension is given explicitly in terms of the volume of the complex torus C d /Γ (see for examples [10,5,13]). ...
We introduce and study the space of -likewise theta functions on with respect to given discrete subgroup and character of . A concrete description is given and an orthonormal basis is then constructed. Its range by the classical Segal-Bargmann transform is also characterized and leads to the so-called thetaBargmann
Fock space.
... Γ is of maximal rank r = 2g or equivalently V 2g R /Γ being compact) is well studied in the literature [17,26,3] and the corresponding space F 2,ν,H Γ 2g ,χ (V) has a high interest not only on its own, but also in the light of the remarkable implications for both pure mathematics and mathematical physics. It is closely connected to number theory and abelian varieties [25,23,5,18], representation theory [10,22], spectral analysis [15,11,12], cryptography and coding theory [21,24], chaoticity of a shift operator [13,14] and quantum field theory [6,8]. Under the cocycle (Riemann-Dirac quantization (RDQ)) condition ...
We are interested in the -holomorphic automorphic functions on a
g-dimensional complex space endowed with a positive
definite hermitian form and associated to isotropic discrete subgroups
of rank . We show that they form an infinite reproducing kernel
Hilbert space which looks like a tensor product of a theta Fock-Bargmann space
on and the classical
Fock-Bargmann space on . Moreover, we provide an explicit
orthonormal basis using Fourier series and we give the expression of its
reproducing kernel function in terms of Riemann theta function of several
variables with special characteristics.
Për mendimin tim, Kalkulusi duhet bërë në shkollën e mesme. Të paktën për ata nxënës që duan të vazhdojnë në degët e shkencave ose inxhinjerive. Leksionet e mëposhtme kanë plot gabime nëpër to sepse nuk i kanë kaluar provës në klasë nga qindra nxënës dhe mësues. Megjithate, për mendimin tim, janë të vetmet në Shqip që synojnë të pregatisin nxënësit me atë bagazh matematike që konsiderohet standart për studentët e shkencave dhe inxhinjerive. Ata mësues që duan ti përdorin këto leksione janë të mirëpritur ta bëjnë këtë dhe kur gjejnë gabime nëpër to ju lutem të më thonë që ti korrigjojmë.
Për mendimin tim, Kalkulusi 3 duhet bërë në shkollën e mesme. Të paktën për ata nxënës që duan të vazhdojnë në degët e shkencave ose inxhinjerive.
Leksionet e mëposhtme kanë plot gabime nëpër to sepse nuk i kanë kaluar provës në klasë nga qindra nxënës dhe mësues. Megjithate, për mendimin tim, janë të vetmet në Shqip që synojnë të pregatisin nxënësit me atë bagazh matematike që konsiderohet standart për studentët e shkencave dhe inxhinjerive.
Ata mësues që duan ti përdorin këto leksione janë të mirëpritur ta bëjnë këtë dhe kur gjejnë gabime nëpër to ju lutem të më thonë që ti korrigjojmë.
The purpose of this paper is to study a further connection between linear codes over three kinds of finite rings and Hermitian lattices over a complex quadratic field (Formula presented.), where l >0 is a square free integer such that l≡3(mod4). Shaska et al. (Finite Fields Appl 16(2): 75–87, 2010) consider a ring R = OKpOK (p is a prime) and study Hermitian lattices constructed from codes over the ring R. We consider a more general ring (Formula presented.), where e ≥1. Using pe allows us to make a connection from a code to a much larger family of lattices. That is, we are not restricted to those lattices whose minimum norm is less than p. We first show that R is isomorphic to one of the following three non-isomorphic rings: a Galois ring GR pe, 2)(Formula presented.), and (Formula presented.). We then prove that the theta functions of the Hermitian lattices constructed from codes over these three rings are determined by the complete weight enumerators of those codes. We show that self-dual codes over R produce unimodular Hermitian lattices. We also discuss the existence of Hermitian self-dual codes over R. Furthermore, we present MacWilliams’ relations for codes over R.
Kurbat algjebrike janë nje nga objektet me të rëndësishme të matematikës. Disa nga problemet me tradicionale të matematikës janë pikërisht mbi kurbat algjebrike. Gjithashtu, me zhvillimin e informatikës dhe mënyrave te reja të llogaritjeve kurbat algjebrike kanë marrë nje rëndësi të vecantë. Ato tani përdoren gjerësisht në kryptografi, teorinë e kodeve dhe mjaft fusha të tjera. Në këtë libër unë jam përpjekur të jap nje trajtim të shkurtër dhe konciz të kasaj dege të gjeometrisë algjebrike.
This paper is the first version of a project of classifying all superelliptic
curves of genus according to their automorphism group. We determine
the parametric equations in each family, the corresponding signature of the
group, the dimension of the family, and the inclussion among such families. At
a later stage it will be determined the decomposition of the Jacobians and each
locus in the moduli spaces of curves.
Ky liber eshte nje hyrje ne algjebren abstrakte per studentët e matematikës. Teksti supozohet të përdoret si tekst bazë puer studentuet e matematikës gjatë një periudhe prej dy semestrash.
This paper is the first version of a project of classifying all su-perelliptic curves of genus g ≤ 48 according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.
This short note describes some of the papers and activities which took place during the NATO Advanced Study Institute in Vlora, Albania during April 28 - May 9, 2008.
Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.
Let X be an algebraic curve of genus g > 2 defined over a field Fq of characteristic p > 0. From X , under certain conditions, we can construct an algebraic geometry code C. If the code C is self-orthogonal under the symplectic product then we can construct a quantum codeQ, called a QAG-code. In this paper we study the construction of such codes from curves with automorphisms and the relation between the automorphism group of the curve X and the codes C and Q.
We study curves and their applications to coding theory. Recently,
Joyner and Ksir have suggested a decoding algorithm based on the automorphisms
of the code. We show how curves can be used to construct MDS codes
and focus on some curves with extra automorphisms, namely
. The automorphism groups of such codes are
determined in most characteristics.
Let \chi be a character on a discrete subgroup \Gamma of rank one of the
additive group (C,+). We construct a complete orthonormal basis of the Hilbert
space of (L^2,\Gamma,\chi)-theta functions. Furthermore, we show that it
possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces
of the Landau operator \Delta_\nu; \nu>0, associated to the eigenvalues \nu m.
For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire
(L^2,\Gamma,\chi)-theta functions. Corresponding orthonormal basis are
constructed and the corresponding reproducing kernel can be expressed in terms
of the generalized theta function of characteristic [\alpha,0].
Ky libër eshte nje hyrje ne algjebren abstrakte per studentet e degeve te matematikes. Mund te perdoret me sukses edhe per studentet e degeve te tjera si shkencave kompjuterike, inxhinjerise elektrike, teorise se komunikacineve, etj.
Algjebra abstrakte eshte jo vetem nje nga deget me te bukura dhe me klasike te matematikes, por edhe nje nga deget me aktive te saj. Sot algjebra abstrakte ka aplikime ne fusha te ndryshme te shkences, industrise, dhe ekonomise, si per shembull në teorine e komunikimeve dixhitale, teorine e kodeve, kriptografi, sistemet financiare, robotike, pervec aplikimeve te fiziken teorike, kimi, biomatematike, etj.
Libri është ndarë në katër pjesë, në teorinë e grupeve, unazave, moduleve, dhe teorinë e fushave.
We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over GF(4) and F 2 × F 2, and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly I modular lattice Z + √ℓZ. We identify some of the hermitian lattices constructed and observe an interesting pair of non-isomorphic 3/2 dimensional codes over F 2 × F 2 that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets GF(4) and F 2 × F 2 are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.
This paper studies the relationship between error-correcting codes over GF(4) and complex lattices (more precisely, [ω]-modules in n, where ). The theta-functions of self-dual lattices are characterized. Two general methods are presented for constructing lattices from codes. Several examples are given, including a new lattice sphere-packing in 36.
A quadratic Jacobi identity to the septic base is introduced and proved by means of modular lattices and codes over rings. As an application the theta series of all the 6-dimensional 7-modular lattices with an Hermitian structure over Q(−7) are derived.
Sphere packings, lattices and groups, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
- J H Conway
- N J A Sloane
J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Second,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 290, Springer-Verlag, New York, 1993.