Conference Paper

# Binary Subblock Energy-Constrained Codes: Knuth’s Balancing and Sequence Replacement Techniques

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## Abstract

The subblock energy-constrained codes (SECCs) have recently attracted attention due to various applications in communication systems such as simultaneous energy and information transfer. In a SECC, each codeword is divided into smaller subblocks, and every subblock is constrained to carry sufficient energy. In this work, we study SECCs under more general constraints, namely bounded SECCs and sliding-window constrained codes (SWCCs), and propose two methods to construct such codes with low redundancy and linear-time complexity, based on Knuth’s balancing technique and sequence replacement technique. For certain codes parameters, our methods incur only one redundant bit.

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... IV. EFFICIENT ENCODERS/DECODERS FOR B RC (n; p) AND Bal RC (n; ǫ) VIA THE SEQUENCE REPLACEMENT TECHNIQUE The Sequence Replacement Technique (SRT) has been widely applied in the literature (for example, see [23], [24], [28], [29]). This is an efficient method for removing forbidden substrings from a source word. ...
... In general, the encoder removes the forbidden strings and subsequently inserts its representation (which also includes the position of the substring) at predefined positions in the sequence. In our recent work [23], for codewords of length m, we enforced the almost-balanced weight-constraint over every window of size ℓ = Ω(log m) (here, a window of size ℓ of x refer ℓ consecutive bits in x). In this section, we show that, for sufficiently large n, the redundancy to encode (decode) binary data to (from) B RC (n; p) when p > 1/2 and Bal RC (n; ǫ) for arbitrary ǫ ∈ (0, 1/2) can be further reduced from Θ(n) bits to only a single bit via the SRT. ...
... • The first encoder adapts the SRT (presented in [23]) with the antipodal matching (constructed in [11]) to encode arbitrary data to B RC (n; p) when p > 1/2 with at most n + 3 redundant bits. • The second encoder, which is the main contribution of this work, modifies the SRT as presented in [23] to encode Bal RC (n; ǫ) with only one redundant bit. ...
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In this work, we study two types of constraints on two-dimensional binary arrays. In particular, given $p,\epsilon>0$, we study (i) The $p$-bounded constraint: a binary vector of size $m$ is said to be $p$-bounded if its weight is at most $pm$, and (ii) The $\epsilon$-balanced constraint: a binary vector of size $m$ is said to be $\epsilon$-balanced if its weight is within $[(0.5-\epsilon)*m,(0.5+\epsilon)*m]$. Such constraints are crucial in several data storage systems, those regard the information data as two-dimensional (2D) instead of one-dimensional (1D), such as the crossbar resistive memory arrays and the holographic data storage. In this work, efficient encoding/decoding algorithms are presented for binary arrays so that the weight constraint (either $p$-bounded constraint or $\epsilon$-balanced constraint) is enforced over every row and every column, regarded as 2D row-column (RC) constrained codes; or over every subarray, regarded as 2D subarray constrained codes. While low-complexity designs have been proposed in the literature, mostly focusing on 2D RC constrained codes where $p = 1/2$ and $\epsilon = 0$, this work provides efficient coding methods that work for both 2D RC constrained codes and 2D subarray constrained codes, and more importantly, the methods are applicable for arbitrary values of $p$ and $\epsilon$. Furthermore, for certain values of $p$ and $\epsilon$, we show that, for sufficiently large array size, there exists linear-time encoding/decoding algorithm that incurs at most one redundant bit.
... On the other hand, the second method, based on Knuth's balancing technique, uses at most nµ(n, p) + O(n log n) redundant bits to construct B(n, p) for arbitrary p < 1/2. We review below the antipodal matching (defined in [7]) and the linear-time encoder for weakly-balanced binary codes [18], as they will be used in our proposed 2D constrained coding methods. ...
... In [18], we studied the weakly-balanced constraint that enforces the weight-constraint over every window of size = Ω(log n), and showed that for arbitrary p 1 , p 2 , where 0 p 1 < 1/2 < p 2 1, for sufficient n, there exists a lineartime algorithm to encode binary data to codewords of length n with only 1 redundant bit. Our coding method is based on the sequence replacement technique and the complexity of the algorithm was shown to be linear in codeword's length [18]. ...
... In [18], we studied the weakly-balanced constraint that enforces the weight-constraint over every window of size = Ω(log n), and showed that for arbitrary p 1 , p 2 , where 0 p 1 < 1/2 < p 2 1, for sufficient n, there exists a lineartime algorithm to encode binary data to codewords of length n with only 1 redundant bit. Our coding method is based on the sequence replacement technique and the complexity of the algorithm was shown to be linear in codeword's length [18]. ...
Conference Paper
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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 .
... Such an index t always exists and there is an efficient method to find t. Indeed, in our recent work [23], we solved a generalization of this problem and showed that for any binary word x ∈ {0, 1} n , there exists an index t such that the word y obtained by flipping the first t bits in x satisfies the weight constraint c 1 n ≤ wt(x) ≤ c 2 n, for given ...
... For simplicity, we assume n > 1 throughout this work. Lemma 2 (Followed by Theorem 2 [23]): Let n be even, and > 0. Given a binary sequence x ∈ {0, 1} n , there exists an index t in the set S ,n , such that t is an -balanced index of x. ...
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We propose coding techniques that simultaneously limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given ℓ, ϵ > 0, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy all of the following properties: • Runlength constraint: the maximum homopolymer run in each codeword is at most ℓ, • GC-content constraint: the GC-content of each codeword is within [0.5 - ϵ; 0.5 + ϵ], • Error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. While various combinations of these properties have been considered in the literature, this work provides generalizations of codes constructions that satisfy all the properties with arbitrary parameters of ℓ and ϵ. Furthermore, for practical values of ℓ and ϵ, we show that our encoders achieve higher rates than existing results in the literature and approach capacity. Our methods have low encoding/decoding complexity and limited error propagation.
... In this work, we focus on the binary channel, where on-off keying is employed, and bit 1 (bit 0) denotes the presence (absence) of a high energy The work of Kui Cai and Tuan Thanh Nguyen is supported by Singapore Ministry of Education Academic Research Fund Tier 2 MOE2019-T2-2-123. This article was presented in part at 2020 IEEE International Symposium on Information Theory [1]. ...
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The subblock energy-constrained codes (SECCs) and sliding window-constrained codes (SWCCs) have recently attracted attention due to various applications in communication systems such as simultaneous energy and information transfer. In a SECC, each codeword is divided into smaller non-overlapping windows, called subblocks, and every subblock is constrained to carry sufficient energy. In a SWCC, however, the energy constraint is enforced over every window. In this work, we focus on the binary channel, where sufficient energy is achieved theoretically by using relatively high weight codes, and study the bounded SECCs and bounded SWCCs, where the weight in every window is bounded between a minimum and maximum number. Particularly, we focus on the cases of parameters that there is no rate loss, i.e. the channel capacity is one, and propose two methods to construct capacity-approaching codes with low redundancy and linear-time complexity, based on Knuth’s balancing technique and sequence replacement technique. These methods can be further extended to construct SECCs and SWCCs. For certain codes parameters, our methods incur only one redundant bit.We also impose the minimum distance constraint for error correction capability of the designed codes, which helps to reduce the error propagation during decoding as well.
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In this work, given n,ϵ > 0, two efficient encoding (decoding) methods are presented for mapping arbitrary data to (from) n×n binary arrays in which the weight of every row and every column is within [(1/2–ϵ)n, (1/2+ϵ)n], which is referred to as the ϵ-balanced constraint. The first method combines the divide and conquer algorithm and a modification of the Knuth’s balancing technique, resulting a redundancy of Θ(n) bits. On the other hand, for sufficiently large n, the second method uses the sequence replacement technique, which costs only one redundant bit. The latter method reduces significantly the redundancy of the best known encoder for two-dimensional p-bounded weight constrained codes from (n + 3) bits to a single bit.
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