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On polyalphabetic block codes
Abstract
A polyalphabetic (or mixed) block code is a set of codewords of finite length, where every symbol of a codeword belongs to its own alphabet. In contrast to previous publications we consider a general case, where we do not assume any algebraic structure of the alphabets and the codes. Upper and lower bounds on the cardinality of a polyalphabetic code with given Hamming distance are obtained. Some constructions of polyalphabetic codes are suggested based on known codes. Encoding and decoding of the polyalphabetic codes, obtained in this way, can be done using encoding and decoding algorithms for the mother code. Using this constructions, codes are obtained, that reach the upper Singleton type bound.
... The output of an encoder has restrictions on the values in the partially-stuck-at cells; in the other cells, it can attain all values. So the set of all encoder outputs is a poly-alphabetic code [23]. To be more precise, the following proposition holds. ...
... As a result of Proposition 4, upper bounds on the size of poly-alphabetic codes [23] are also upper bounds on the size of partially-stuck-at codes. Hence, we give the following corollaries to state Singleton-type and sphere-packing-type bounds for error-correcting and partially-stuck-at-masking codes. ...
... Let C u,t be a code by Theorem 8 whose rate is R given in Table 5 at u row and t column. Taking C 19,27 gives C 20,26 and C 23,26 with R = 0.435 applying Theorem 5 and Lemma 1, respectively. In contrary, taking C 19,31 is advantageous as there are codes (C 20,30 by Theorem 5 and C 23,30 by Lemma 1) with R = 0.380 while direct application of Theorem 8 cannot provide these codes as highlighted in green with "None". ...
This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask u partially stuck cells while correcting at the same time t random errors. The process of “masking” determines a word whose entries coincide with writable levels at the (partially) stuck cells. For u>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u>1$$\end{document} and alphabet size q>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>2$$\end{document}, our new constructions improve upon the required redundancy of known constructions for t=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=0$$\end{document}, and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert–Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert–Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction.
... However, in some circumstances this assumption might not hold. Sidorenko et al. [18] suggested that this is the case in orthogonal-frequency-division-multiplexing (OFDM) transmission; it can also be viewed as a relaxation of the partially-stuck-cell setting [1], where both sender and receiver are aware (perhaps thorough a periodic sampling routine) of which coordinates have smaller alphabets. The authors further believe that this generalization of classical error correction is of independent theoretical interest (see, e.g., their study in [5,Ch. ...
... In comparison to [6], [8]- [10], [13], [17], the codes we consider are not necessarily perfect, and therefore can correct more than a single error. On the other hand, [18] generalized the Singleton bound to mixed codes, and presented constructions of MDS codes, based on this bound and known MDS "mother codes", by letting alphabet sizes grow with respect to the code block size. Similarly, [5,Cha. ...
... Then, in Section IV we study the size of Hamming spheres in this space; in Section V we observe the list-decoding capabilities of mixed codes by generalizing the first Johnson bound, and in Section VI we develop lower-and upper bounds on the sizes of mixed codes. Finally, in Section VII we demonstrate that our bounds improve upon the known bound of [18,Th. 2], [5,Cor. ...
Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect codes) or in the case of unbounded alphabet sizes, we focus on the case of finite alphabets, and generalize the Gilbert-Varshamov, sphere-packing, Elias-Bassalygo, and first linear programming bounds to that setting. In the latter case, our proof is also the first for the non-symmetric mono-alphabetic $q$-ary case using Navon and Samorodnitsky's Fourier-analytic approach.
... The output of an encoder has restrictions on the values in the partially stuck-at cells; in the other cells, it can attain all values. So the set of all encoder outputs is a poly-alphabetic code [22]. To be more precise, the following proposition holds. ...
... As a result of Proposition 4, upper bounds on the size of poly-alphabetic codes [22] are also upper bounds on the size of partially-stuck-at codes. ...
This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask $u$ partially stuck cells while correcting at the same time $t$ random errors. The process of "masking" determines a word whose entries coincide with writable levels at the (partially) stuck cells. For $u>1$ and alphabet size $q>2$, our new constructions improve upon the required redundancy of known constructions for $t=0$, and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert--Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert--Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction.
... The rate of an NF-code C = C K with parameters (n, a 1 , . . . , a n ; M, s) is defined to be the quotient R(C) = log |M C |/ n i=1 log Na i (see [39,Section II]). Guruswami [12, Section E] notes that a standard argument from the geometry of numbers suggests that |M C | ≈ (2 r1 π r2 M d )/(d! |D K |). ...
This paper presents a list decoding algorithm for the number field codes of Guruswami (IEEE Trans Inf Theory 49:594–603, 2003). The algorithm is an implementation of the unified framework for list decoding of algebraic codes of Guruswami, Sahai and Sudan (Proceedings of the 41st Annual Symposium on Foundations of Computer Science, 2000), specialised for number field codes. The computational complexity of the algorithm is evaluated in terms of the size of the inputs and field invariants.
... is tight since ǫ is small. Finally, as noted in [4], the codes C ′ u and C ′′ v may be decoded using the decoders for their respective " parent " codes. With that, it is clear that (C) Φ may be decoded using the linear-time decoder in [3,Fig. ...
— Recently, Roth and Skachek proposed two methods for constructing nearly maximum-distance separable (MDS) expander codes. We show that through the simple modification of using mixed-alphabet codes derived from MDS codes as constituent codes in their code designs, one can obtain nearly MDS codes of significantly smaller alphabet size, albeit at the expense of a (very slight) reduction in code rate. Index Terms — Expander codes, linear-time encodable and decodable codes, maximum-distance-separable codes, mixedalphabet codes I.
Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least 3 (thus one-error-correcting) for a fixed redundancy r, base-field size q and field-extension degree m (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is 1, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.
The Chinese remainder codes has been studied when only errors occur. We propose an error-erasure-decoder for the Chinese remainder codes and analyze the decoding radius.
A new class of subcodes in rank metric is proposed; based on it, multicomponent network codes are constructed. Basic properties of subspace subcodes are considered for the family of rank codes with maximum rank distance (MRD codes). It is shown that nonuniformly restricted rank subcodes reach the Singleton bound in a number of cases. For the construction of multicomponent codes, balanced incomplete block designs and matrices in row-reduced echelon form are used. A decoding algorithm for these network codes is proposed. Examples of codes with seven and thirteen components are given.
In this paper two fundamental aspects for wireless data transmission are addressed, namely the interrelation between multicarrier spread spec- trum transmission and multilevel coding and the view of downlink trans- mission as the broadcast channel from information theory. In the first part it will be shown that multicarrier spread spectrum systems can be interpreted as a multilevel code with repetition codes in all levels and different mappings onto the modulation alphabet. This point of view might provide new insights into both, multicarrier spread spec- trum systems as well as multilevel coding. In the second part we will describe the information theory point of view for the situation when data are transmitted from one sender to many receivers. This is called the broadcast channel. The opposite case when data are transmitted from many senders to one receiver is called the multiple access chan- nel. It is known that the broadcast channel has a larger capacity than the multiple access channel which might be exploited in the downlink transmission .W ith the help of elementary examples these relations will be illustrated.
We consider distance profiles of arbitrary block codes (over arbitrary alphabets) with unequal error correcting capabilities. We show that for group codes (and hence for codes linear over fields) these distance profiles can be obtained using well known MaxLog a posteriori probability decoding algorithms.
The paper reveals the interrelation between multicarrier spread spectrum (MC-SS) transmission and multilevel coding (MLC). It is shown that MC-SS systems can be interpreted as a multilevel code with repetition codes in the levels. The paper is intended to provide new insights into both MC-SS systems and MLC. For both schemes, a lot of theory has been provided by numerous publications and particular parts of the existing theory for one scheme can be applied to the other.
Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following “curve-fitting” problem over a field F: Given n points {(x<sub>i</sub>.y<sub>i</sub>)}<sub>i=1</sub> <sup>n</sup>, x<sub>i</sub>,y<sub>i</sub>∈F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y<sub>i</sub>=p(x<sub>i</sub>) for all but at most e values of i∈{1....,n}. We give an algorithm that solves this problem for e<n-√(kn), which improves over the previous best result, for every choice of k and n. Of particular interest is the case of k/n>1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometric codes. In both cases, we obtain a list decoding algorithm that corrects up to n-√(n-d-) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometric codes extends the methods of Shokrollahi and Wasserman (1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve fitting problem, which is of use in soft-decision decoding algorithms for Reed-Solomon codes
Upper and lower bounds are presented for the maximal possible size
of mixed binary/ternary error-correcting codes. A table up to length 13
is included. The upper bounds are obtained by applying the linear
programming bound to the product of two association schemes. The lower
bounds arise from a number of different constructions
A construction for an infinite family of perfect mixed codes with
covering radius 2 is presented. These are the first known nontrivial
perfect mixed codes with covering radius greater than 1. Based on mixed
codes, constructions for binary covering codes that lead to a
considerable improvement of upper bounds on the sizes of covering codes
are presented. These codes and some other codes can be obtained by the
blockwise direct sum construction. Two infinite families of codes are of
special interest. They are quasi-perfect, nonlinear, union of their
disjoint translates covers the space, and their density as covering
codes is remarkably low
The need to transmit and store massive amounts of data reliably and without error is a vital part of modern communications systems. Error-correcting codes play a fundamental role in minimising data corruption caused by defects such as noise, interference, crosstalk and packet loss. This book provides an accessible introduction to the basic elements of algebraic codes, and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book was first published in 2003 and is aimed at graduate students of electrical and computer engineering, and at practising engineers whose work involves communications or signal processing.
Vectors (x1, x2, …, xn) are considered with components xi being elements of groups Gi which may vary with i. Linear and nonlinear single-error-correcting perfect codes are constructed in this framework by generalizing algebraic methods of Zaremba.
Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following “curve-fitting” problem over a field F: given n points ((x<sub>i</sub>·y<sub>i</sub>))<sub>i=1 </sub><sup>n</sup>, x<sub>i</sub>, y<sub>i</sub>∈F, and a degree parameter k and error parameter e, find all univariate polynomials p of degree at most k such that y<sub>i</sub>=p(x<sub>i</sub>) for all but at most e values of i∈(1,...,n). We give an algorithm that solves this problem for e<n-√(kn), which improves over the previous best result, for every choice of k and n. Of particular interest is the case of k/n>1/3, where the result yields the first asymptotic improvement in four decades. The algorithm generalizes to solve the list decoding problem for other algebraic codes, specifically alternant codes (a class of codes including BCH codes) and algebraic-geometry codes. In both cases, we obtain a list decoding algorithm that corrects up to n-√(n(n-d')) errors, where n is the block length and d' is the designed distance of the code. The improvement for the case of algebraic-geometry codes extends the methods of Shokrollahi and Wasserman (see in Proc. 29th Annu. ACM Symp. Theory of Computing, p.241-48, 1998) and improves upon their bound for every choice of n and d'. We also present some other consequences of our algorithm including a solution to a weighted curve-fitting problem, which may be of use in soft-decision decoding algorithms for Reed-Solomon codes
Polyalphabetic codes, Ninth Int. Workshop on Algebraic and Combinatorial Coding Theory
- E Gabidulin
- M Bossert
E. Gabidulin, M. Bossert, Polyalphabetic codes, Ninth Int. Workshop on Algebraic and Combinatorial Coding Theory, Kranevo, Bulgaria, June 2004, pp. 171-177.