Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these can be very simply as images under Gray map of over Z/sub 4/, integers mod 4 (although this requires a slight modification codes). The construction implies are distance invariant. Duality in 4/ domain have dual weight...

A new algebraic approach to quasi-cyclic codes is introduced. The key idea regard a code over field as linear an auxiliary ring. By the use of Chinese remainder theorem (CRT), or discrete Fourier transform (DFT), that ring can be decomposed into direct product fields. That decomposition in turn yields construction from lower lengths which turns out some cases celebrated squaring and cubing constructions other (u+/spl upsi/|u-/spl upsi/) Vandermonde constructions. All binary extended...

Le critère de Robin spécifie que l'hypothèse Riemann (RH) est vraie si et seulement l'inégalité σ(n):=∑ d|n d<e γ nloglogn vérifiée pour n≥5041, avec la constante d'Euler(-Mascheroni). Nous montrons par des méthodes élémentaires n≥37 ne satisfait pas au il doit être pair n'est ni sans facteur carré non divisible exactement un premier. Utilisant une borne Rosser Schoenfeld, nous montrons, en outre, n puissance cinquième >1. Comme corollaire, obtenons RH ssi chaque entier naturel >1 vérifie Robin.

We consider the Gaussian wiretap channel, where two legitimate players Alice and Bob communicate over an additive white noise (AWGN) while Eve is eavesdropping, also through AWGN channel. propose a coding strategy based on lattice coset encoding. define secrecy gain as design criterion for codes, expressed in terms of theta series, which characterizes Eve's confusion function channel parameters. The studied even unimodular lattices, asymptotic analysis shows that it grows exponentially...

The alphabet F/sub 2/+uF/sub 2/ is viewed here as a quotient of the Gaussian integers by ideal (2). Self-dual codes with Lee weights multiple 4 are called Type II. They give even unimodular lattices Construction A, while I yield lattices. B makes it possible to realize Leech lattice lattice. There Gray map which maps II into binary fixed point free involution in their automorphism group. Combinatorial constructions use weighing matrices and strongly regular graphs. Gleason-type theorems for...

Introduces a new image coding scheme using lattice vector quantization. The proposed method involves two steps: biorthogonal wavelet transform of the image, and quantization coefficients. In order to obtain compromise between minimum distortion bit rate, we must truncate scale suitably. To meet this goal, need know how many points lie within truncated area. We investigate case Laplacian sources where surfaces equal probability are spheres for L/sup 1/ metric (pyramids) arbitrary lattices....

We construct new self-dual and isodual codes over the integers module 4. The binary images of these under Gray map are nonlinear, but formally self-dual. construction involves Hensel lifting cyclic codes. Quaternary quadratic residue obtained by classical Repeated produces a universal code defined 2-adic integers. investigate connections between this Z/sub 4/, composition automorphism group, structure idempotents 4/. also derive square root bound on minimum Lee weight, explore with finite...

We generalize the construction of linear codes via skew polynomial rings by using Galois instead finite fields as coefficients. The resulting non commutative are no longer left and right Euclidean. Codes that principal ideals in quotient a two sided studied. As an application, constacyclic self-dual over $GR(4, 2)$ constructed. Euclidean give $\mathbb Z_4$−codes. Hermitian yield 3−modular lattices quasi-cyclic

Linear complementary dual (LCD) codes are binary linear that meet their trivially. We construct LCD using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from over the family of rings Rk. give a programming bound on largest size an code given length minimum distance. make table lower bounds for this function modest values parameters.

Linear complementary dual (LCD) codes are binary linear that meet their trivially. We construct LCD using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from over the family of rings Rk. give a programming bound on largest size an code given length minimum distance. make table lower bounds for this function modest values parameters.

Type II Z/sub 4/-codes are introduced as self-dual codes over the integers modulo 4 containing all-one vector and with Euclidean weights multiple of 8. Their weight enumerators characterized by means invariant theory. A notion extremality for is introduced. binary images under Gray map formally even weights. Extended quadratic residue main example this family codes. They obtained Hensel lifting classical have good parameters. With every type 4/-code associated via construction an unimodular...

The Nordstrom-Robinson, Kerdock, and (slightly modified) Preparata codes are shown to be linear over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z 4"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathbb {Z}_4}</mml:annotation> </mml:semantics> </mml:math>...

Following Parts I and II, quasi-cyclic codes of given index are studied as over a finite polynomial ring.These latter decomposed by the Chinese Remainder Theorem (CRT), or equivalently Mattson-Solomon transform, into products shorter larger alphabets.We characterize enumerate self-dual one-generator quasicyclic in that context.We give an algorithm to remove some equivalent from enumeration.A generalization multigenerator is sketched.

We construct an infinite family of two-Lee-weight codes over the ring $\mathbb {F}_{2}+u\mathbb {F}_{2}$ . These are defined as trace codes. They have algebraic structure abelian Their Lee weight distribution is computed by using character sums. By Gray mapping, we obtain binary two-weight shown to be optimal application Griesmer bound. An secret sharing schemes given.