Lee Distance Research Articles

Asymptotic properties of Lee distance

Distances on permutations are often convenient tools for analyzing and modeling rank data. They measure the closeness between two rankings and can be very useful and informative for revealing the main structure and features of the data. In this paper, some statistical properties of the Lee distance are studied. Asymptotic results for the random variable induced by Lee distance are derived and used to compare the Distance-based probability model and the Marginals model for complete rankings. Three rank datasets are analyzed as an illustration of the presented models.

Negacyclic codes over the local ring [formula omitted] of oddly even length and their Gray images

Let R=Z4[v]/〈v2+2v〉=Z4+vZ4 (v2=2v) and n be an odd positive integer. Then R is a local non-principal ideal ring of 16 elements and there is a Z4-linear Gray map from R onto Z42 which preserves Lee distance and orthogonality. First, a canonical form decomposition and the structure for any negacyclic code over R of length 2n are presented. From this decomposition, a complete classification of all these codes is obtained. Then the cardinality and the dual code for each of these codes are given, and self-dual negacyclic codes over R of length 2n are presented. Moreover, all 23⋅(4p+5⋅2p+9)2p−2p negacyclic codes over R of length 2Mp and all 3⋅(4p+5⋅2p+9)2p−1−1p self-dual codes among them are presented precisely, where Mp=2p−1 is a Mersenne prime. Finally, 36 new and good self-dual 2-quasi-twisted linear codes over Z4 with basic parameters (28,228,dL=8,dE=12) and of type 21447 and basic parameters (28,228,dL=6,dE=12) and of type 21646 which are Gray images of self-dual negacyclic codes over R of length 14 are listed.

Two-weight and three-weight codes from trace codes over [formula omitted

We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring Fp+uFp+vFp+uvFp, where u2=0,v2=0,uv=vu. These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over Fp. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.

Asymmetric Lee Distance Codes for DNA-Based Storage

We introduce a new family of codes, termed asymmetric Lee distance (ALD) codes, designed to correct errors arising in DNA-based storage systems and systems with parallel string transmission protocols. ALD codes are defined over a quaternary alphabet and analyzed in this particular setting, but the derived results hold for other alphabet sizes as well. Our technical contributions are twofold. First, we derive upper bounds on the size of the codes under the ALD metric based on linear programming techniques. Second, we propose a number of code constructions, which imply lower bounds.

Complex Hadamard Matrix-Aided Generalized Space Shift Keying Modulation

In this paper, we present a complex Hadamard matrix-aided generalized space shift keying (HSSK) modulation scheme, which introduces complex-Hadamard-based signal vectors at the transmitter to provide higher spectrum efficiency than generalized space shift keying (GSSK). It also has lower maximum-likelihood detection complexity than the multiple active spatial modulation (MA-SM) and generalized spatial modulation (GSM) systems due to the corresponding fast complex Hadamard transform at the receiver. Based on the Lee distance of our signal vectors, we provide an optimized mapping between our data bits and signal vectors. We also analyze the upper bound on the average bit error rate of our HSSK system, which agrees well with the simulation results. Finally, the performance of HSSK is compared with MA-SM, GSM, and GSSK systems through Monte Carlo simulations. It is shown that for the same transmission rate the HSSK system performs better than the GSM, MA-SM, and GSSK systems.

A class of skew-constacyclic codes over ℤ<SUB align="right">4 + uℤ<SUB align="right">4

In this paper, we study a class of skew-constacyclic codes over R = Z4 + uZ4, which is a non-chain extension of Z4. Some structural properties of R[x, θ] are discussed, where θ is an automorphism of R. We determine a necessary condition and a sufficient condition for these codes to be free, when they are principally generated. A Gray map over R is defined and some good codes are obtained using it. For even n, a relation between the generator polynomial of a code and that of its dual is obtained. Some examples are given to illustrate the results. Further, we have generalised these codes to double skew-constacyclic codes over R. Some good codes with improved minimum Lee distance over Z4 have been found via this class, and the same have been added to the database of Z4 codes.

Minimum Lee-moment aberration and its applications

Various criteria have been proposed to compare and assess fractional factorial designs. Among them the minimum moment aberration criterion is conceptually simple and computationally cheap. However, the power moments in the minimum moment aberration criterion depend on the Hamming distances between two runs, which leads to minimum moment aberration criterion having a few defects. In this paper, we use Lee distance instead of Hamming distance to de ne power moments, and propose the minimum Lee-moment aberration criterion to compare and assess multi-level fractional factorial designs. Some relationships between minimum Lee-moment aberration criterion and other criteria are also studied, which provides a signi cant statistical justi cation of minimum Lee-moment aberration criterion.

On the lifted Zetterberg code

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial $$g(x)=(x+1)p(x)$$g(x)=(x+1)p(x), where p(x) is the minimum polynomial over GF(2) of an element of order $$2^m+1$$2m+1 in $$GF(2^{2m})$$GF(22m) and m is even. This even binary code has parameters $$[2^m+1,2^m-2m, 6]$$[2m+1,2m-2m,6]. The quaternary code obtained by lifting the code to the alphabet $${\mathbb {Z}}_4=\{0,1,2,3\}$$Z4={0,1,2,3} is shown to have parameters $$[2^m+1,2^m-2m, d_L ]$$[2m+1,2m-2m,dL], where $$d_L \ge 8$$dL?8 denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters $$(2^{m+1}+2,2^k,d_H)$$(2m+1+2,2k,dH), where $$d_H \ge 8$$dH?8 denotes the minimum Hamming weight and $$k=2^{m+1}-4m$$k=2m+1-4m. For $$m=6$$m=6 these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these $${\mathbb {Z}}_4$$Z4-codes for all even m. This appears to be the first infinite family of $${\mathbb {Z}}_4$$Z4-codes of length $$n=2^m+1$$n=2m+1 with $$d_L \ge 8$$dL?8 having an algebraic decoding algorithm.

On the lifted Melas code

The binary Melas code is a cyclic code with generator polynomial g(u)=p(u)p(u)? where p(u) is a primitive polynomial of odd degree m?5 and the ? denotes reciprocation. The even-weight subcode of a Melas code has generator polynomial (u+1)g(u) and parameters [2m?1,2m?2m?2,6]. This code is lifted to ?4$\mathbb {Z}_{4}$ and the quaternary code is shown to have parameters [2m?1,2m?2m?2,dL?8], where dL denotes the minimum Lee distance. An algebraic decoding algorithm correcting all errors of Lee weight ≤3 is presented for this code. The Gray map of this quaternary code is a binary code with parameters [2m+1?2,2m+1?4m?4,dH?8] where dH is the minimum Hamming distance. For m=5,7 the minimum distance equals the minimum distance of the best known linear code for the given length and code size.

Lower bounds on the minimum distance of long codes in the Lee metric

The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over $$\mathbb F _p$$ or $$\mathbb F _{p^2},$$ or over even degree extensions of $$\mathbb F _p.$$ In the latter case the approach is concatenation with a good code for the Hamming metric as outer code and a short code for the Lee metric as an inner code. In the former case lower bounds on the minimum Lee distance are derived by algebraic geometric arguments inspired by results of Wu et al. (Electron Lett 15(43):820–821, 2007).

On Maximum Lee Distance Codes

Singleton-type upper bounds on the minimum Lee distance of general (not necessarily linear) Lee codes over ℤq are discussed. Two bounds known for linear codes are shown to also hold in the general case, and several new bounds are established. Codes meeting these bounds are investigated and in some cases characterised.

Energy Minimization of Repelling Particles on a Toric Grid

We explore the minimum energy configurations of repelling particles distributed over $n$ possible locations forming a toric grid. We conjecture that the most energy-efficient way to distribute $n/2$ particles over this space is to place them in a checkerboard pattern. Numerical experiments validate this conjecture for reasonable choices of the repelling force. In the present paper, we prove this conjecture in a large number of special cases---most notably, when the sizes of the torus are either two or multiples of four in all dimensions and the repelling force is a completely monotonic function of the Lee distance between the particles.

Minimum Weights and Weight Enumerators of $\BBZ_{4}$-Linear Quadratic Residue Codes

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z4 is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z4-linear XQR-code of length 80 is a new example of an extremal Z4-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.

Algebraic decoding of negacyclic codes over $${\mathbb Z_4}$$

In this article we investigate Berlekamp's negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp's original papers: our codes have minimum Lee distance at least 2t + 1, where the generator polynomial of the code has roots ?, ? 3, . . . , ? 2t-1 for a primitive 2nth root ? of unity in a Galois extension of $${\mathbb {Z}_4}$$ ; no restriction on t is imposed. We present an algebraic decoding algorithm for this class of codes that corrects any error pattern of Lee weight ? t. Our treatment uses Grobner bases, the decoding complexity is quadratic in t.

A note on the minimum Lee distance of certain self-dual modular codes

In a former paper we investigated the connection between $p$-ary linear codes, $p$ prime, and theta functions. Corresponding to a given code a suitable lattice and its associated theta function were defined. Using results from the theory of modular forms we got an algorithm to determine an upper bound for the minimum Lee distance of certain self-dual codes. In this note we generalize this result to $m$-ary codes, where $m$ is either a power of a prime, or $m$ is square-free. If $m$ is of a different form the generalization will not work. A class of examples to illustrate this fact is given.

The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes

Let C m be the cycle of length m . We denote the Cartesian product of n copies of C m by G ( n , m ) : = C m □ C m □ ⋯ □ C m . The k -distance chromatic number χ k ( G ) of a graph G is χ ( G k ) where G k is the k th power of the graph G = ( V , E ) in which two distinct vertices are adjacent in G k if and only if their distance in G is at most k . The k -distance chromatic number of G ( n , m ) is related to optimal codes over the ring of integers modulo m with minimum Lee distance k + 1 . In this paper, we consider χ 2 ( G ( n , m ) ) for n = 3 and m ≥ 3 . In particular, we compute exact values of χ 2 ( G ( 3 , m ) ) for 3 ≤ m ≤ 8 and m = 4 k , and upper bounds for m = 3 k or m = 5 k , for any positive integer k . We also show that the maximal size of a code in Z 6 3 with minimum Lee distance 3 is 26.

A $\mathbb Z$<SUB><I>4</I></SUB>-linear code of high minimum Lee distance derived from a hyperoval

In this paper we present a new non-free $\mathbb Z$<SUB><I>4</I></SUB>-linear code of length $29$ and size $128$ whose minimum Lee distance is $28$. Its Gray image is a nonlinear binary code with parameters $(58,2^7,28)$, having twice as many codewords as the biggest linear binary codes of equal length and minimum distance. The code also improves the known lower bound on the maximal size of binary block codes of length $58$ and minimum distance $28$. <br>&nbsp;&nbsp; Originally the code was found by a heuristic computer search. We give a geometric construction based on a hyperoval in the projective Hjelmslev plane over $\mathbb Z$<SUB><I>4</I></SUB> which allows an easy computation of the symmetrized weight enumerator and the automorphism group. Furthermore, a generalization of this construction to all Galois rings of characteristic $4$ is discussed.

Uniform Design for Experiments with Mixtures

The goal of uniform mixture design is to scatter the design points in the experimental region uniformly. The commonly used criteria, such as mean square distance, are based on the Euclidean distance. Based on the Lee distance, a new criterion is proposed in this article. And an algorithm, called NTLBG, is also proposed to refine the randomly generated design for the experimental design with mixtures. Some examples show that the design generated by the NTLBG algorithm has a lower criteria value.

Key predistribution using combinatorial designs for grid-group deployment scheme in wireless sensor networks

We propose a new grid-group deployment scheme in wireless sensor networks. We use combinatorial designs for key predistribution in sensor nodes. The deployment region is divided into square regions. The predistribution scheme has the advantage that all nodes within a particular region can communicate with each other directly and nodes which lie in a different regions can communicate via special nodes called agents which have more resources than the general nodes. The number of agents in a region is always three, whatever the size of the network. We give measures of resiliency taking the Lee distance into account. Apart from considering the resiliency in terms of fraction of links broken, we also consider the resiliency as the number of nodes and regions disconnected when some sensor are compromised. This second measure, though very important, had not been studied so far in key predistribution schemes which use deployment knowledge. We find that the resiliency as the fraction of links compromised is better than existing schemes. The number of keys preloaded in each sensor node is much less than all existing schemes and nodes are either directly connected or connected via two hop paths. The deterministic key predistribution schemes result in constant-time computation overhead for shared key discovery and path key establishment.

On the distances of cyclic codes of length [formula omitted] over [formula omitted

Cyclic codes of length 2 e over Z 4 are precisely the ideals of the local ring Z 4 [ x ] / 〈 x 2 e − 1 〉 . In this paper, we investigate the distances of cyclic codes of length 2 e over Z 4 . The Hamming distances of all cyclic codes of length 2 e over Z 4 are determined. The exact Lee distances of some cyclic codes of length 2 e over Z 4 are also obtained.