Permutation Codes Research Articles

Scheduling optimization of aging employees considering physical load

Considering the influence of age on the task completion time of the workforce, the scheduling problem of the workforce under physical load was extended, and a mathematical optimization model was proposed, preventing the workforce from continuously engaging in high workload work in the same body part and minimizing the standard deviation of task completion time. The influence of age on the workforce's reaction and endurance ability was discussed and formulated reasonably. The high and low workload of the workstation was evaluated by Rapid entire body assessment. A particle swarm optimization algorithm was designed by permutation coding based on set rules, and the rationality of the model and the effectiveness of the algorithm were proved by an example. The results show that the model can not only eliminate the risk of continuous high workload but also improve the performance of the assembly line.

Notice of Removal: A Novel Representation for Permutations

In this paper, we present a variable-length binary code for permutations of degree n, where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover's enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary n in a straightforward way.

Injection Coding with Spectral Nulls for Even Codeword Lengths

With a view to adapting information signals to suit the frequency characteristics of digital transmission channels, it is pertinent to design channel coding schemes that can exhibit spectral shaping capabilities after being processed with a suitable modulation technique. This involves concentrating the frequency energy of the coded and modulated information signal towards a predetermined range of the frequency spectrum, or making it have low power content at such frequencies. One modulation scheme that is suitable for achieving this is pulse amplitude modulation (PAM). We thus present a permutation coding (PC) system with injections that can exhibit spectral nulls at rational sub-multiples of the symbol frequency. Such injections are achieved by strategically removing δ columns from the source PC. We also present a mathematical expression that allows for the prediction of zero energy positions in the codebook’s spectrum. Due to the way the injections are introduced, it gives the scheme an advantage of achieving higher symbol rate, when compared with conventional PC systems. The work is however limited to cases where δ =2 and the lengths of the codewords involved are even numbers.

Injection Coding With Spectral Shaping Capability

In digital transmission systems that involve pulse amplitude modulation (PAM), channel coding is useful for spectral shaping by concentrating the frequency energy of the transmitted information towards a predetermined range of the frequency spectrum, or making it have low power content at such frequencies, so as to suit the frequency characteristics of the communication channel. We thus present a permutation coding system with injections that can exhibit spectral shaping at rational sub-multiples of the symbol frequency. We also present a mathematical expression that allows for the prediction of low energy positions in the codebook's spectrum. Upper and lower bounds for the low energy spectrum were also determined. Due to the way the injections are introduced, it gives the scheme an advantage of achieving higher symbol rate, when compared with conventional permutation coding systems.

On the cyclic order distribution and partitioning of linear cyclic codes

In this paper, we propose a way to partition any linear cyclic code in its cyclic equivalence classes according to the algebraic structure of the code. Such partition is useful to generate cyclically permutable codes directly from linear cyclic codes. We consider the case where the size of the finite field and the block length of the code are coprimes. Moreover, we present a criterion to construct maximal cyclic order codes.

Rate-Improved Permutation Codes for Correcting a Single Burst of Deletions

Permutation codes are widely studied due to their promising applications in flash memories. Based on the theory of permutation groups and subgroups, two classes of permutation codes are constructed to correct a single burst deletion of length up to a designated parameter. The proposed codes can achieve larger rates than available codes while maintaining simple interleaving structures. The decoding methods for the proposed codes are provided in proofs and verified by examples.

Combinatorial generation via permutation languages. I. Fundamentals

In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain four classical Gray codes for permutations, bitstrings, binary trees and set partitions as special cases. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into $n$ rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group $S_n$. Recently, Pilaud and Santos realized all those lattice congruences as $(n-1)$-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.

Generalized Path–Permutation Codes for Reliable End-to-End Networking of Opportunistic Links

Opportunistic links are emerging as a novel technology for state-of-the-art wireless networks, such as cognitive radio networks, energy-harvesting networks, and low-latency vehicular networks. To increase the reliability of these networks, path-permutation codes (PPCs) were advocated in [1] . These codes are based on a virtual multiple-input multiple-output (MIMO) model on the session/network layer, and have been tailored for increased reliability of opportunistic links. By exploiting multipath–multihop routes, PPCs access one relay path at a time while repeatedly transmitting the same QAM symbol. In this paper, we introduce a generalized PPC scheme which simultaneously accesses multiple relay paths and comprises multiple QAM symbols in the packet. Phase rotations are also introduced in the PPC codewords for additional performance improvement. Numerical simulation results validate our analyses and demonstrate the superior performance of this novel coding scheme, i.e., better error rate performance, higher throughput, and stronger robustness to transmission outages.

Novel low complexity decoding method for the binary coded frequency shift keying modulation

SummaryThis paper proposes a coded modulation scheme based on frequency shift keying (FSK) modulation and a simple binary permutation code. FSK detection is based on the knowledge of the probability density function (PDF) of the noise and the attenuation of the channel, called FSK “Soft‐Decision” (FSK/SD) detection. We introduce an approximation of the FSK/SD, called FSK “Select Largest” (FSK/SL). We compare the FSK/SL detection with the classical PDF‐based detection techniques, such as FSK/SD and spread‐FSK (SFSK), for both coded and uncoded transmission. As a practical application, we use the power line communication (PLC) channel. We analyze and compare the performances for a narrowband PLC channel with attenuation and impulse noise.

Multi-Permutation Codes Correcting a Single Burst Unstable Deletions in Flash Memory

Multi-permutation codes for rank modulation scheme have been recently proposed for efficiently storing data in flash memories. In this letter, we investigate the problem of a single burst unstable deletion (BUD) of length $t$ in a $t$ -balanced multi-permutation and this problem can be decomposed into the sub-problems of a single permutation-invariant deletion (PID) in $t$ permutations by a rank demodulation method. Based on the rank demodulation method and the interleaving of $t$ single-PID-correcting permutation codes, we construct two classes of $t$ -balanced multi-permutation codes that are capable of correcting a single BUD of length $t$ . Moreover, the constructed codes can achieve higher rates compared with the existing code. The decoding methods are provided in the proofs and verified by some examples.

Permutation Coding with Injections for Modified PAM System

Arriving at a good combination of coding and modulation schemes that can achieve good error correction constitutes a challenge in digital communication systems. In this work, we explore the combination of permutation coding (PC) and pulse amplitude modulation (PAM) for mitigating channel errors in the presence of background noise and jitter. Since PAM is characterised with bi-polar constellations, Euclidean distance is a good choice for predicting the performance of such coded modulation setup. In order to address certain challenges facing PCs, we therefore introduce injections in the coding system, together with a modified form of PAM system. This modification entails constraining the PAM constellations to the size of the codeword’s symbol. The results obtained demonstrate the strength of the modified coded PAM system over the conventional PC coded PAM system.

The Construction of Permutation Group Codes for Communication Systems: Prime or Prime Power?

This paper focuses on correcting a theorem relating to the construction of permutation group codes (PGCs). The theorem in question assumed that affine transformation could be used to enumerate all the code words of a permutation array with a minimum Hamming distance (MHD) of n - 1 for any n > 1. This assumption was founded upon the proposition that, if the code length, n, is a prime power, then the maximum cardinality of the code will be n(n - 1) and its MHD will be n - 1. However, two typical algebraic methods, one relying on affine transformation and another upon the composite operation of two small subgroups of a symmetric group, can violate this proposition. This is because it is only when n is a prime rather than a prime power that they can enumerate all the code words of an (n; n (n - 1) ; n - 1)-PGC. By investigating how the range of n impacts upon the cardinality and MHD of a code, we provide a corrective theorem that stipulates the construction of (p q ; p 2q (1 - 1/p); p q (1 - 1/p))-PGCs when n = pq is a prime power, wherep is a prime and q > 1. On the basis of this theorem, and under the condition of n being the power of 2, we construct a (2 q ; 2 2q-1 ; 2 q-1 )-PGC and present an encoder that can map a k-bit binary information sequence to an n = 2 2q -dimension permutation code word. The natural array structure of (2 q ; 2 2q-1 ; 2 q-1 )-PGCs makes them especially well-suited to forming the basis of low-complexity encoders. We present simulation-based experiments that show that, as the code length increases, the performance of these codes improves. The best performance is for a (16; 128; 8)-PGC, which can achieve -3.8 dB with a word error rate of 10 -7 .

New Lower Bounds for Permutation Codes Using Linear Block Codes

In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an [n,k,d] <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree n, having minimum distance at least d and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.

Ulam Ball Size Analysis for Permutation and Multipermutation Codes Correcting Translocation Errors

Permutation and multipermutation codes in the Ulam metric have been suggested for use in non-volatile memory storage systems such as flash memory devices. In this paper we introduce a new method to calculate permutation ball sizes in the Ulam metric using Young Tableaux and prove the non-existence of non-trivial perfect permutation codes in the Ulam metric. We then extend the study to multipermutations, providing upper and lower bounds on multipermutation Ulam ball sizes and resulting upper and lower bounds on the maximal size of multipermutation codes in the Ulam metric.

A Performance Evaluation and Two New Implementations of Evolutionary Algorithms for Land Partitioning Problem

Many bio-inspired techniques are proposed and implemented to solve real-world applications. The number of these techniques is increasing day by day, so the researchers (especially out of computer sciences) have difficulty in deciding which technique to select for the problem. In this study, two new implementations to solve land partitioning problem and also a performance analysis of three evolutionary algorithms were carried out on this real-world engineering problem. Land partitioning is a discrete optimization problem that cannot be solved in linear time with conventional techniques. Two new implementations of automated land partitioning (ALP-DE and ALP-SS) were carried out by using differential evolution algorithm (DE) and scatter search (SS) methods. The algorithms were adapted to the land partitioning problem by being discretized with permutation coding. These two proposed methods were compared with a similar study in the published literature and a designer’s plan for a project area that contains 18 blocks using a mathematical model. These proposed automatic methods (ALP-DE and ALP-SS) resulted in more successful and more appropriate partitioning plans than those of a designer in accordance with land partitioning criteria. When the comparison of these three different evolutionary algorithms was examined, the ALP-SS method showed superior performance in all blocks. The low standard deviation values of the proposed methods indicated that both methods are robust and successful tools for the land partitioning problem.

Constructing permutation arrays using partition and extension

We give new lower bounds for M(n, d), for various positive integers n and d with $$n>d$$, where M(n, d) is the largest number of permutations on n symbols with pairwise Hamming distance at least d. Large sets of permutations on n symbols with pairwise Hamming distance d are needed for constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, partition and extension, is universally applicable to constructing such sets for all n and all d, $$d<n$$. We describe three new techniques, sequential partition and extension, parallel partition and extension, and a modified Kronecker product operation, which extend the applicability of partition and extension in different ways. We describe how partition and extension gives improved lower bounds for $$M(n,n-1)$$ using mutually orthogonal Latin squares (MOLS). We present efficient algorithms for computing new partitions: an iterative greedy algorithm and an algorithm based on integer linear programming. These algorithms yield partitions of positions (or symbols) used as input to our partition and extension techniques. We report many new lower bounds for M(n, d) found using these techniques for n up to 600.

MOCZ for Blind Short-Packet Communication: Practical Aspects

We investigate practical aspects of a recently introduced blind (noncoherent) communication scheme, called modulation on conjugate-reciprocal zeros (MOCZ). MOCZ is suitable for a reliable transmission of sporadic and short-packets at ultra-low latency and high spectral efficiency via unknown multipath channels, which are assumed to be static over the receive duration of one packet. The information is modulated on the zeros of the transmitted discrete-time baseband signal's z- transform. Because of ubiquitous impairments between the transmitter and receiver clocks, a carrier frequency offset occurs after down-conversion to the baseband. This results in a common rotation of the zeros. To identify fractional rotations of the base angle in the zero-pattern, we propose an oversampled direct zero-testing decoder to identify the most likely one. Integer rotations correspond to cyclic shifts of the binary message, which we determine by cyclically permutable codes (CPC). Additionally, the embedding of CPCs into cyclic codes, enables additive error-correction which reduces the bit-error-rate tremendously. Furthermore, we exploit the trident structure in the signal's autocorrelation for an energy based detector to estimate timing offsets and the effective channel delay spread. We finally demonstrate how this joint data and channel estimation can be largely improved by receive antenna diversity at low SNR.

Burst-Deletion-Correcting Codes for Permutations and Multipermutations

Permutation codes and multipermutation codes are widely studied due to various applications in information theory. Designing codes correcting deletion errors has been the main subject of works in the literature and to the best of our knowledge, there exist only optimal codes capable of correcting a single deletion in a permutation. In this paper, we construct several classes of permutation and multipermutation codes that are capable of correcting a burst deletion of length s ≥ 2, for both stable and unstable models. Efficient error decoders are provided to show the correctness of our constructions.

A lower bound on permutation codes of distance $$n-1$$

A classical recursive construction for mutually orthogonal latin squares (MOLS) is shown to hold more generally for a class of permutation codes of length n and minimum distance $$n-1$$. When such codes of length $$p+1$$ are included as ingredients, we obtain a general lower bound $$M(n,n-1) \ge n^{1.0797}$$ for large n, gaining a small improvement on the guarantee given from MOLS.

Doubly Resolvable Canonical Kirkman Packing Designs and its Applications

Let $$v \equiv 4 \pmod 6$$ and $$v\ge 4$$ . A canonical Kirkman packing design of order v, denoted by $$\hbox {CKPD}(v)$$ , is a resolvable packing with $$r=(v-4)/2$$ parallel classes such that (i) each parallel class consists of a size 4 block and $$(v- 4)/3$$ triples; (ii) the leave consists of the union of $$(v- 4)/2$$ vertex-disjoint edges and a $$K_4$$ with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable $$\hbox {CKPD}(v)\hbox {s}$$ with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained.