A generalization of the binary Preparata code

AbstractA classical binary Preparata code P2(m) is a nonlinear (2m+1,22(2m-1-m),6)-code, where m is odd. It has a linear representation over the ring Z4 [Hammons et al., The Z4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319]. Here for any q=2l>2 and any m such that (m,q-1)=1 a nonlinear code Pq(m) over the field F=GF(q) with parameters (q(Δ+1),q2(Δ-m),d⩾3q), where Δ=(qm-1)/(q-1), is constructed. If d=3q this set of parameters generalizes that of P2(m). The equality d=3q is established in the following cases: (1) for a series of initial admissible values q and m such that qm<2100; (2) for m=3,4 and any admissible q, and (3) for admissible q and m such that there exists a number m1 with m1|m and d(Pq(m1))=3q. We apply the approach of [Nechaev and Kuzmin, Linearly presentable codes, Proceedings of the 1996 IEEE International Symposium Information Theory and Application Victoria, BC, Canada 1996, pp. 31–34] the code P is a Reed–Solomon representation of a linear over the Galois ring R=GR(q2,4) code P dual to a linear code K with parameters near to those of generalized linear Kerdock code over R

Propelinear structure of Z_{2k}-linear codes

Let C be an additive subgroup of $\Z_{2k}^n$ for any $k\geq 1$. We define a Gray map $\Phi:\Z_{2k}^n \longrightarrow \Z_2^{kn}$ such that \Phi(\codi) is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover, $\Phi$ is the unique Gray map such that $\Phi(C)$ is Hamming-compatible group code. Using this Gray map we discuss about the nonexistence of 1-perfect binary mixed group code

On Completely Regular Codes

This work is a survey on completely regular codes. Known properties, relations with other combinatorial structures and constructions are stated. The existence problem is also discussed and known results for some particular cases are established. In particular, we present a few new results on completely regular codes with covering radius 2 and on extended completely regular codes

Z2Z4-linear codes: generator matrices and duality

A code ${\cal C}$ is $\Z_2\Z_4$-additive if the set of coordinates can be partitioned into two subsets $X$ and $Y$ such that the punctured code of ${\cal C}$ by deleting the coordinates outside $X$ (respectively, $Y$) is a binary linear code (respectively, a quaternary linear code). In this paper $\Z_2\Z_4$-additive codes are studied. Their corresponding binary images, via the Gray map, are $\Z_2\Z_4$-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for $\Z_2\Z_4$-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of $\Z_2\Z_4$-additive codes are given.Comment: This paper will be submitted to IEEE Trans. on Inform. Theor

Decoder for Nonbinary CWS Quantum Codes

We present a decoder for nonbinary CWS quantum codes using the structure of union codes. The decoder runs in two steps: first we use a union of stabilizer codes to detect a sequence of errors, and second we build a new code, called union code, that allows to correct the errors

A new approach to codeword stabilized quantum codes using the algebraic structure of modules

In this work, we study the Codeword Stabilized Quantum Codes (CWS codes) a generalization of the stabilizers quantum codes using a new approach, the algebraic structure of modules, a generalization of linear spaces. We show then a new result that relates CWS codes with stabilizer codes generalizing results in the literature.Comment: 7 page

On Codes Over Zps\mathbb{Z}_{p^{s}} with the Extended Lee Weight

We consider codes over $\mathbb{Z}_{p^s}$ with the extended Lee weight. We find Singleton bounds with respect to this weight and define MLDS and MLDR codes accordingly. We also consider the kernels of these codes and the notion of independence of vectors in this space. We investigate the linearity and duality of the Gray images of codes over $\mathbb{Z}_{p^s}$

On Codes over Zp2\mathbb{Z}_{p^2} and its Covering Radius

This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{p^2}$ with respect to Lee distance. We also determine the covering radius of various Repetition codes over $\mathbb{Z}_{p^2}.

On the number of nonequivalent propelinear extended perfect codes

The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation

The Geometry of Deep Networks: Power Diagram Subdivision

We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be {\em max-affine spline operators} (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partitioning corresponds to a {\em power diagram} (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the (exponentially many) power diagram regions to greatly reduce their complexity. For classification problems, we obtain a formula for a MASO DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's nonlinearities, weights, and architecture. Numerous numerical experiments support and extend our theoretical results