Distance-preserving Mappings Research Articles

Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem

Abstract It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.

New Distance Concept and Graph Theory Approach for Certain Coding Techniques Design and Analysis

AbstractA New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from thek-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in thek-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze.

Nonbinary convolutional codes and modified M-FSK detectors for power-line communications channel

The Viterbi decoding algorithm, which provides maximum ¿ likelihood decoding, is currently considered the most widely used technique for the decoding of codes having a state description, including the class of linear error-correcting convolutional codes. Two classes of nonbinary convolutional codes are presented. Distance preserving mapping convolutional codes and M-ary convolutional codes are designed, respectively, from the distance-preserving mappings technique and the implementation of the conventional convolutional codes in Galois fields of order higher than two. We also investigated the performance of these codes when combined with a multiple frequency-shift keying (M-FSK) modulation scheme to correct narrowband interference (NBI) in powerline communications channel. The modification of certain detectors of the M-FSK demodulator to refine the selection and the detection at the decoder is also presented. M-FSK detectors used in our simulations are discussed, and their chosen values are justified. Interesting and promising obtained results have shown a very strong link between the designed codes and the selected detector for M-FSK modulation. An important improvement in gain for certain values of the modified detectors was also observed. The paper also shows that the newly designed codes outperform the conventional convolutional codes in a NBI environment.

A Comparative Study on the Distance-Optimality for Distance-Preserving Mappings Constructions with Cyclic-Shift Prefix Technique

This paper presents a comparative study on the distance optimality for Distance-Preserving Mappings codes generated from different constructions. Few constructions are considered to investigate the reasons behind making codes optimum when they reach the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences. The technique of cyclic-shift prefix or suffix of permutation symbols is used to investigate the conditions making distance-preserving mappings codes optimum.

Constructing Constant Composition Codes via Distance-Increasing Mappings

A distance-preserving mapping is a one-to-one function $f$ from $p$-ary vectors of length $m$ to $q$-ary vectors of length $n$ such that any two distinct $p$-ary vectors are mapped to two different $q$-ary vectors with an equal or greater Hamming distance. A distance-increasing mapping is a special distance-preserving mapping which strictly increases the distance by at least one if the distance of two distinct input vectors is less than the length of the output vectors. A constant composition code over a $k$-ary alphabet has the property that the numbers of occurrences of the $k$ symbols within a codeword are fixed for each codeword. One of the most important applications of distance-preserving mappings and distance-increasing mappings is to construct constant composition codes, of which the permutation codes are a special subclass. There are two results in this paper. First, we propose a swap-based distance-increasing mapping from binary vectors to quaternary constant composition vectors. Second, we prove that it is impossible to construct any swap-based distance-preserving mappings from binary vectors to ternary constant composition vectors under the swap model that we defined.

Permutation sequences and coded PAM signals with spectral nulls at rational submultiples of the symbol frequency

Coded PAM signals with spectral nulls at rational submultiples of the symbol frequency are presented. Spectral shaping using permutation symbols and distance-preserving mappings (DPMs) are two techniques presented in this paper to design codes with better error correction capability, which make them achieve a significant decoding gain compared to other published codes. The use of Viterbi decoding algorithm, gives these new codes an advantage with their property of no error propagation. The well shaped power spectral densities of these new codes may overcome some communications problem like zero frequency components.

Simple Distance-Preserving Mappings From Ternary Vectors to Permutations

We give a simple construction of distance-preserving mappings from ternary vectors to permutations (3-DPM). Our result gives a lower bound for permutation arrays, i.e., P(n, d) ges A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> (n, d) , which significantly improves previous lower bounds for d les 3n / 5.

Distance-Preserving and Distance-Increasing Mappings From Ternary Vectors to Permutations

Permutation arrays have found applications in powerline communication. One construction method for permutation arrays is to map good codes to permutations using a distance-preserving mappings (DPM). DPMs are mappings from the set of all q-ary vectors of a fixed length to the set of permutations of some fixed length (the same or longer) such that every two distinct vectors are mapped to permutations with the same or larger Hamming distance than that of the vectors. A DPM is called distance increasing (DIM) if the distances are strictly increased (except when the two vectors are equal). In this correspondence, we propose constructions of DPMs and DIMs from ternary vectors. The constructed DPMs and DIMs improve many lower bounds on the maximal size of permutation arrays.

Using Graphs for the Analysis and Construction of Permutation Distance-Preserving Mappings

A new way of looking at permutation distance-preserving mappings (DPMs) is presented by making use of a graph representation. The properties necessary to make such a graph distance-preserving, are also investigated. Further, this new knowledge is used to analyze previous constructions, as well as to construct a new general mapping algorithm for a previous multilevel construction.

A Generalized Upper Bound and a Multilevel Construction for Distance-Preserving Mappings

A new general upper bound is derived on the sum of the Hamming distances between sequences when mapping from one set of sequences to another. It is shown that a similar upper bound for mappings from binary sequences to permutation sequences is a special case of this upper bound and this is used to evaluate known mappings. Also, new distance-preserving mappings (DPMs) from binary sequences to permutation sequences are presented, based on a multilevel construction. In addition to explicit distance-conserving mappings, distance-increasing, and distance-reducing mappings are also presented. Several of the new DPMs attain the upper bound

Permutation Trellis Codes

We introduce the new concept of permutation trellis codes and present a generalized construction procedure, applying our technique of distance-preserving mappings. Minimum-distance decoding follows naturally, using the Viterbi algorithm. We furthermore investigate the performance of these codes when combined with multitone frequency-shift keying modulation and noncoherent detection in a diversity scheme, to make transmissions robust against narrowband, broadband, and background noise disturbances, such as those encountered in power-line communications.

Distance-increasing mappings from binary vectors to permutations

Mappings from the set of binary vectors of a fixed length to the set of permutations of the same length that strictly increase Hamming distances except when that is obviously not possible are useful for the construction of permutation codes. In this correspondence, we propose recursive and explicit constructions of such mappings. Some comparisons show that the new mappings have better distance expansion distributions than other known distance-preserving mappings (DPMs). We also give some examples to illustrate the applications of these mappings to permutation arrays (PAs).

Distance-preserving mappings from binary vectors to permutations

Mappings of the set of binary vectors of a fixed length to the set of permutations of the same length are useful for the construction of permutation codes. In this article, several explicit constructions of such mappings preserving or increasing the Hamming distance are given. Some applications are given to illustrate the usefulness of the construction. In particular, a new lower bound on the maximal size of permutation arrays (PAs) is given.

On the Mazur-Ulam Theorem and the Aleksandrov Problem for Unit Distance Preserving Mappings

Let $X$ and $Y$ be two real normed vector spaces. A mapping $f:X \to Y$ preserves unit distance in both directions iff for all $x,y \in X$ with $||x - y|| = 1$ it follows that $||f(x) - f(y)|| = 1$ and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve unit distance in both directions. We shall prove that such mappings are not very far from being isometries. This problem was asked by A. D. Aleksandrov. The first classical result that characterizes isometries between normed real vector spaces goes back to S. Mazur and S. Ulam in 1932. We also obtain an extension of the Mazur-Ulam theorem.