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Constant Weight Codes: An Approach Based on Knuth's Balancing Method

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Abstract

In this article, we study properties and algorithms for constructing sets of 'constant weight' codewords with bipolar symbols, where the sum of the symbols is a constant q, q 6 0. We show various code constructions that extend Knuth's balancing vector scheme, q = 0, to the case where q > 0. We compute the redundancy of the new coding methods. Index Terms—Balanced code, channel capacity, constrained code, magnetic recording, optical recording. I. INTRODUCTION Let q be an integer. A setC, which is a subset of ( w = (w1;w2;:::;wn)2f 1; +1g n : n X i=1 wi = q )

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... In fact, this problem was posed by MacWilliams and Sloane as Research Problem 17.3 [15]. To the best of our knowledge, there are three encoding approaches: the enumerative method of Schalkwijk [16], the geometric approach of Tian et al. [17], and the Knuth-like method of Skachek and Immink [18]. For the case where q is constant, the first two methods encode in quadratic time O(n 2 ), while the third method runs in linear time. ...
... That is, there exist binary words x with T (x, q) = ∅ and we refer to such words as bad words. Hence, a different encoding rule must be applied to these bad words, and simple linear-time methods were proposed and studied by Skachek and Immink [18]. While their q-balancing schemes are in fact variable-length schemes, Skachek and Immink did not provide an analysis of the average redundancy of their schemes and instead argued that log 2 n+O(1) redundant bits are sufficient in the worst case (when q is constant). ...
... (I) In this paper, we amalgamate the variable-length scheme in [10] with the q-balancing schemes in [18] to obtain new variable-length q-balancing schemes. We formally describe Schemes A and B in Sections III and V, respectively. ...
Preprint
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We study and propose schemes that map messages onto constant-weight codewords using variable-length prefixes. We provide polynomial-time computable formulas that estimate the average number of redundant bits incurred by our schemes. In addition to the exact formulas, we also perform an asymptotic analysis and demonstrate that our scheme uses $\frac12 \log n+O(1)$ redundant bits to encode messages into length-$n$ words with weight $(n/2)+{\sf q}$ for constant ${\sf q}$.
... However, the construction is limited to specific constant dimension codes. In [2], a construction of CW codes based on Knuth's balancing approach [3] is presented. The flexibility on the tail bits is used to generate CW codes including balanced codes. ...
... In the previous section, the construction of q-ary CW sequences was achieved with a weight range shown in (2). However, because of the limited interval, we will present an approach to extend this range. ...
... Example 4: Consider the same ternary information sequence x = 212 of length 3 as in Example 3. We would like to generate a (7, 3, 12, 3) CW sequence of weight W = 12 and n = 7 as described in Table II. We observe from (2) that the information sequence 212 can only generate (n, k, W, q) CW sequences with W ∈ [2,10]. In order to extend this range of weight, we append a ternary redundant vector u of length e to c . ...
Article
We present an encoding and decoding scheme for constant weight sequences, that is, given an information sequence, the construction results in a sequence of specific weight within a certain range. The scheme uses a prefix design that is based on Gray codes. Furthermore, by adding redundant symbols we extend the range of weight values for output sequences, which is useful for some applications.
... However, the construction is limited to specific constant dimension codes. In [2], a construction of CW codes based on Knuth's balancing approach [3] is presented. The flexibility on the tail bits is used to generate CW codes including balanced codes. ...
... In the previous section, the construction of q-ary CW sequences was achieved with a weight range shown in (2). However, because of the limited interval, we will present an approach to extend this range. ...
... Example 4: Consider the same ternary information sequence x = 212 of length 3 as in Example 3. We would like to generate a (7, 3, 12, 3) CW sequence of weight W = 12 and n = 7 as described in Table II. We observe from (2) that the information sequence 212 can only generate (n, k, W, q) CW sequences with W ∈ [2,10]. In order to extend this range of weight, we append a ternary redundant vector u of length e to c . ...
Conference Paper
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We present an encoding and decoding scheme for constant weight sequences, that is, given an information sequence, the construction results in a sequence of specific weight within a certain range. The scheme uses a prefix design that is based on Gray codes. Furthermore, by adding redundant symbols we extend the range of weight values for output sequences, which is useful for some applications.
... • Step 1: Calculating probabilities for programming the memory cells to the LRS; • Step 2: Encoding row-by-row the user message M of length-into A , such that every row and column satisfies the corresponding weight-constraints. For the first row, we can use any coding methods for constructing the 1-D constant-weight codes [9]- [11]. In this work, we apply the enumerative coding technique [10] so as to achieve a high coding efficiency. ...
... Due to 2 < 2 ( 2 = 140), then 2 = 2 . The index sets 1 and 0 where 1 = (1, 6, 9, 10) (filled by the yellow color) and 0 = (2,3,4,5,7,8,11) (no fill) are then assigned by C 1 ( , 2,1 →1 ) and C 0 ( , 2,0 →1 ), respectively. Two codewords are then extracted from the codebook P 2 . ...
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This paper proposes novel methods for designing two-dimensional (2-D) weight-constrained codes for reducing the parasitic currents in the crossbar resistive memory array. In particular, we present efficient encoding/decoding algorithms for capacity-approaching 2-D weight-constrained codes of size m×n, where each row has a weight pn with p < 1/2; and each column has a weight qm with q ≤ 1/2. We show that the designed codes provide higher code rates compared to the prior art codes for p ≤ 1/2 and q ≤ 1/2.
... The main reason is due to the provable difficulty of 2D-constraints compared to 1D-constraints. For example, consider certain weight-constrained codes such as the balanced codes or constant-weight codes, there are several efficient prior-art coding methods for designing 1Dcodes with optimal or almost optimal redundancy [14]- [17]. Here, almost optimal refers to the cases that the encoder's redundancy is at most a constant bit away from the optimal redundancy. ...
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In this work, given n, p>0 , efficient encoding/decoding algorithms are presented for mapping arbitrary data to and from n×n binary arrays in which the weight of every row and every column is at most pn. Such constraint, referred as p-bounded-weight-constraint, is crucial for reducing the parasitic currents in the crossbar resistive memory arrays, and has also been proposed for certain applications of the holographic data storage. While low-complexity designs have been proposed in the literature for only the case p=1/2 , this work provides efficient coding methods that work for arbitrary values of p . The coding rate of our proposed encoder approaches the channel capacity for all p .
... Encoding and decoding of constant composition code is a field of active research, see, for example [4], [5], [6]. For the binary case, Weber and Immink [7] and Skachek et al. [8] presented methods that translate arbitrary data into a codeword having a prescribed number of one's and zero's. Enumerative methods for generating codewords have been presented in [9], [10], [11]. ...
... Given that balanced sequences are a specic case of constant-weight sequences, Knuth's algorithm was extended to constant-weight sequences in [19]. Other algorithms for constructing constant-weight sequences can be found in [28,34,52]. ...
Thesis
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Our research deals with the encoding and decoding of balanced sequences using Gray codes. Given that any non-binary sequence, can always be balanced through certain algorithms, we show that the encoding and decoding of a balanced sequence can be performed through a simple and efficient method where the prefix is a Gray code. Our balancing scheme makes use of a generalization of Knuth's balancing algorithm, per- formed on the overall sequence length which includes the information sequence as well as the designed prefix. Our proposed method was firstly applied to certain information source lengths and then generalized for any length. We conclude with a detailed complexity and redundancy analysis for our balancing algorithm.
... In [9], Carlet determined one weight linear codes over Z 4 and in [23], Wood studied linear one weight codes over Z m . Constant weight codes are very useful in a variety of applications such as data storage, fault-tolerant circuit design and computing, pattern generation for circuit testing, identification coding, and optical overlay networks [20]. Moreover, the reader can find the other applications of constant weight codes; determining the zero error decision feedback capacity of discrete memoryless channels in [21], multiple access communications and spherical codes for modulation in [14,15], DNA codes in [18,19], powerline communications and frequency hopping in [11]. ...
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Inspired by the Z2Z4-additive codes, linear codes over Z2^r x(Z2+uZ2)^s have been introduced by Aydogdu et al. more recently. Although these family of codes are similar to each other, linear codes over Z2^r x(Z2+uZ2)^s have some advantages compared to Z2Z4-additive codes. A code is called constant weight(one weight) if all the codewords have the same weight. It is well known that constant weight or one weight codes have many important applications. In this paper, we study the structure of one weight Z2Z2[u]-linear and cyclic codes. We classify these type of one weight codes and also give some illustrative examples.
... The mapping of bits into activation patterns and vice-versa can be seen as a constant weight coding that maps between unconstrained binary b 1 -tuples and constant weight binary n-tuples of weight k. A lot of work has been done on constant weight coding [31]- [33], with special focus on balanced codes (k = n/2) [34]- [36]. In the original OFDM-IM scheme [24], a mapping that is based on the combinatorial number system of degree k [37, p. 27-30] was employed. ...
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A balanced code with r check bits and k information bits is a binary code of length k+r and cardinality 2<sup>k</sup> such that each codeword is balanced; that is, it has [(k+r)/2] 1's and [(k+r)/2] 0's. This paper contains new methods to construct efficient balanced codes. To design a balanced code, an information word with a low number of 1's or 0's is compressed and then balanced using the saved space. On the other hand, an information word having almost the same number of 1's and 0's is encoded using the single maps defined by Knuth's (1986) complementation method. Three different constructions are presented. Balanced codes with r check bits and k information bits with k&les;2<sup>r+1</sup>-2, k&les;3×2<sup>r</sup>-8, and k&les;5×2<sup>r</sup>-10r+c(r), c(r)∈{-15, -10, -5, 0, +5}, are given, improving the constructions found in the literature. In some cases, the first two constructions have a parallel coding scheme
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In a balanced code each codeword contains equally many 1's and 0's. Parallel decoding balanced codes with 2<sup>r</sup> (or 2<sup>r </sup>-1) information bits are presented, where r is the number of check bits. The 2<sup>2</sup>-r-1 construction given by D.E. Knuth (ibid., vol.32, no.1, p.51-3, 1986) is improved. The new codes are shown to be optimal when Knuth's complementation method is used
Article
For n >0, d &ges;0, n ≡ d (mod 2), let K ( n , d ) denote the minimal cardinality of a family V of ±1 vectors of dimension n , such that for any ±1 vector w of dimension n there is a v ∈ V such that | v - w |&les; d , where v - w is the usual scalar product of v and w . A generalization of a simple construction due to D.E. Knuth (1986) shows that K ( n , d )&les;[ n /( d +1)]. A linear algebra proof is given here that this construction is optimal, so that K ( n , d )-[ n /( d +1)] for all n ≡ d (mod 2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links
Article
A constant weight, w, code with k information bits and r check bits is a binary code of length n=k+r and cardinality 2<sup>k</sup> such that the number of 1s in each code word is equal to w. When w=[n/2], the code is called balanced. This paper describes the design of balanced and constant weight codes with parallel encoding and parallel decoding. Infinite families of efficient constant weight codes are given with the parameters k, r, and the “number of balancing functions used in the code design,” ρ. The larger ρ grows, the smaller r will be; and the codes can be encoded and decoded with VLSI circuits whose sizes and depths are proportional to pk and log<sub>2</sub> p, respectively. For example, a design is given for a constant weight w=33 code with k=64 information bits, r=10 check bits, and p=8 balancing functions. This code can be implemented by a VLSI circuit using less than 4,054 transistors with a depth of less than 30 transistors
degrees in computer science from the Technion—Israel Institute of Technology he visited the Mathematics of Communications Department at Bell Laboratories under the DIMACS Special Focus Program in Computational Information Theory and Coding. During
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M.Sc. and Ph.D. degrees in computer science from the Technion—Israel Institute of Technology, in 1994, 1998 and 2007, respectively. In the summer of 2004, he visited the Mathematics of Communications Department at Bell Laboratories under the DIMACS Special Focus Program in Computational Information Theory and Coding. During 2007-2012, he held research positions with the Claude Shannon Institute, University College Dublin, with the School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, and with the Department of Electrical and Computer Engineering, McGill University, Montréal. He is now a Senior Lecturer with the Institute of Computer Science, University of Tartu.
Skachek is a recipient of the Permanent Excellent Faculty Instructor award, given by Technion
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Dr. Skachek is a recipient of the Permanent Excellent Faculty Instructor award, given by Technion.
System-on-chip test architectures: nanometer design for testability
  • L.-T Wang
  • C E Stroud
  • N A Touba
L.-T. Wang, C.E. Stroud, and N.A. Touba, "System-on-chip test architectures: nanometer design for testability," Elsevier Science Limited, 2008.