Symbol-pair Read Channels Research Articles

Function-Correcting Codes for Symbol-Pair Read Channels

Function-Correcting Codes for Symbol-Pair Read Channels

The b-symbol weight hierarchy of the Kasami codes

The symbol-pair read channel was first proposed by Cassuto and Blaum. Later, Yaakobi et al. generalized it to the b-symbol read channel. It is motivated by the limitations of the reading process in high density data storage systems. One main task in b-symbol coding theory is to determine the b-symbol weight hierarchy of codes. In this paper, we study the b-symbol weight hierarchy of the Kasami codes, which are well known for their applications to construct sequences with optimal correlation magnitudes. The complete symbol-pair weight distribution of the Kasami codes is determined.

On Linear Perfect b-Symbol Codes over Finite Fields

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where b≥2 in 2016. In this paper, we establish a b-sphere-packing bound and present a recurrence relationship for the b-weight enumerator. We determine all parameters of linear perfect b-symbol e-error-correcting codes over Fq for e<2b and show that for 2b≤e<3b, there exist at most finite such codes for a given b, e, and q. We construct a family of linear perfect b-symbol b-error-correcting codes over Fq using constacyclic codes.

MDS and AMDS symbol-pair codes constructed from repeated-root cyclic codes

Symbol-pair codes introduced by Cassuto and Blaum in 2010 are designed to protect against the pair errors in symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. Based on repeated-root cyclic codes, we construct two classes of MDS symbol-pair codes for more general generator polynomials and also give a new class of almost MDS (AMDS) symbol-pair codes with the length lp. In addition, we derive all MDS and AMDS symbol-pair codes with length 3p, when the degree of the generator polynomials is no more than 10. The main results are obtained by determining the solutions of certain equations over finite fields.

AMDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance is of significance in determining the error-correcting capability of a symbol-pair code. One of the central themes in symbol-pair coding theory is the constructions of symbol-pair codes with the largest possible minimum symbol-pair distance. Maximum distance separable (MDS) and almost maximum distance separable (AMDS) symbol-pair codes are optimal and sub-optimal regarding the Singleton bound, respectively. In this paper, six new classes of AMDS symbol-pair codes are explicitly constructed through repeated-root cyclic codes. Remarkably, one class of such codes has unbounded lengths and the minimum symbol-pair distance of another class can reach 13.

Constructions of MDS symbol-pair codes with minimum distance seven or eight

Symbol-pair codes are proposed to guard against pair-errors in symbol-pair read channels. The minimum symbol-pair distance plays a vital role in determining the error-correcting capability and the constructions of symbol-pair codes with largest possible minimum symbol-pair distance is of great importance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that such codes can acheive the Singleton bound. In this paper, for length 5p, two new classes of MDS symbol-pair codes with minimum symbol-pair distance seven or eight are constructed by utilizing repeated-root cyclic codes over $${\mathbb {F}}_{p}$$ , where p is a prime. In addition, we derive a class of MDS symbol-pair codes with minimum symbol-pair distance seven and length 4p.

MDS symbol-pair codes from repeated-root cyclic codes

Symbol-pair codes are proposed to protect against pair-errors in symbol-pair read channels. The research of symbol-pair codes with the largest possible minimum symbol-pair distance is of great significance since such codes have the best error-correcting capability. A symbol-pair code meeting the Singleton bound is called a maximum distance separable (MDS) symbol-pair code. In this paper, two new classes of MDS symbol-pair codes are proposed by utilizing repeated-root cyclic codes over finite fields with odd characteristic. It should be noted that these codes have minimum symbol-pair distance ten or twelve.

Geometric Approach tob-Symbol Hamming Weights of Cyclic Codes

Symbol-pair codes were introduced by Cassuto and Blaum in 2010 to protect pair errors in symbol-pair read channels. Recently Yaakobi, Bruck and Siegel (2016) generalized this notion to b-symbol codes in order to consider consecutive b errors for a prescribed integer b ≥ 2, and they gave constructions and decoding algorithms. Cyclic codes were considered by various authors as candidates for symbol-pair codes and they established minimum distance bounds on (certain) cyclic codes. In this paper we use algebraic curves over finite fields in order to obtain tight lower and upper bounds on b-symbol Hamming weights of arbitrary cyclic codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> . Here b ≥ 2 is an arbitrary prescribed positive integer and \mathbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> is an arbitrary finite field. We also present a stability theorem for an arbitrary cyclic code C of dimension k and length n: the b-symbol Hamming weight enumerator of C is the same as the k-symbol Hamming weight enumerator of C if k ≤ b ≤ n-1. Moreover, we give improved tight lower and upper bounds on b-symbol Hamming weights of some cyclic codes related to irreducible cyclic codes. Throughout the paper the length n is coprime to q.

On symbol-pair weight distribution of MDS codes and simplex codes over finite fields

Symbol-pair code is a new coding framework proposed to combat pair-errors in symbol-pair read channels. Remarkably, a classical maximum distance separable (MDS) code is also an MDS symbol-pair code. In this paper, we investigate the symbol-pair weight distribution of MDS codes and the symbol b-weight distribution of simplex codes over finite fields respectively. Surprisingly, one can derive that all the nonzero codewords of simplex codes have the same symbol b-weight and rearrangement of coordinates in simplex codes may induce different symbol-pair weights. Moreover, the symbol b-weight distribution of variation simplex codes over certain finite fields is determined.

Bounds and Constructions of Codes Over Symbol-Pair Read Channels

Cassuto and Blaum recently studied the symbol-pair channel, a model where every two consecutive symbols are read together. This special channel structure is motivated by the limitations of the reading process in high density data storage systems, where it is no longer possible to read individual symbols. In this new paradigm, the errors are not individual symbol errors, but rather symbol-pair errors , where at least one of the symbols is erroneous. In this work, we study bounds and constructions of codes over the symbol-pair channel. We extend the Johnson bound and the linear programming bound for this channel and show that they improve upon existing bounds. We then propose new code constructions that improve upon existing results for pair-distance six, seven, and ten.

New MDS Symbol-Pair Codes From Repeated-Root Codes

Symbol-pair codes can be used to guard against pair errors over symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this letter, we construct three new MDS symbol-pair codes with minimum pair-distance six or seven through repeated-root constacyclic codes. In contrast with classical MDS codes, they have relatively large length.

Constacyclic Symbol-Pair Codes: Lower Bounds and Optimal Constructions

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai {\it et al.} (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes.

The symbol-pair distance distribution of a class of repeated-root cyclic codes over F p m $\phantom {\dot {i}\!}\mathbb {F}_{p^{m}}$

Symbol-pair codes are proposed to protect against pair errors in symbol-pair read channel. One main task in symbol-pair coding theory is to determine the minimum pair-distance of symbol-pair codes. In this paper, we investigate the symbol-pair distance of cyclic codes of length p e over $\phantom {\dot {i}\!}\mathbb {F}_{p^{m}}$ . The exact symbol-pair distance of all cyclic codes of such length is determined.

Error-Trapping Decoding for Cyclic Codes over Symbol-Pair Read Channels

Error-Trapping Decoding for Cyclic Codes over Symbol-Pair Read Channels

Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pair-distance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.

Constructions and Decoding of Cyclic Codes Over &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$b$ &lt;/tex-math&gt; &lt;/inline-formula&gt;-Symbol Read Channels

Symbol-pair read channels, in which the outputs of the read process are pairs of consecutive symbols, were recently studied by Cassuto and Blaum. This new paradigm is motivated by the limitations of the reading process in high density data storage systems. They studied error correction in this new paradigm, specifically, the relationship between the minimum Hamming distance of an error correcting code and the minimum pair distance, which is the minimum Hamming distance between symbol-pair vectors derived from codewords of the code. It was proved that for a linear cyclic code with minimum Hamming distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> , the corresponding minimum pair distance is at least d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> +3. In this paper, we show that, for a given linear cyclic code with a minimum Hamming distance d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> , the minimum pair distance is at least d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</sub> + (dH/2). We then describe a decoding algorithm, based upon a bounded distance decoder for the cyclic code, whose symbol-pair error correcting capabilities reflect the larger minimum pair distance. Finally, we consider the case where the read channel output is a larger number, b ≥3, of consecutive symbols, and we provide extensions of several concepts, results, and code constructions to this setting.

Linear Programming Decoding of Binary Linear Codes for Symbol-Pair Read Channel

In this paper, we develop a new decoding algorithm of a binary linear codes for symbol-pair read channels. Symbol-pair read channel has recently been introduced by Cassuto and Blaum to model channels with high write resolution but low read resolution. The proposed decoding algorithm is based on a linear programming (LP). It is proved that the proposed LP decoder has the maximum-likelihood (ML) certificate property, i.e., the output of the decoder is guaranteed to be the ML codeword when it is integral. We also introduce the fractional pair distance $d_{fp}$ of a code which is a lower bound on the pair distance. It is proved that the proposed LP decoder will correct up to $\lceil d_{fp}/2\rceil-1$ pair errors.

Algebraic Decoding of BCH Codes over Symbol-Pair Read Channels: Cases of Two-Pair and Three-Pair Error Correction

Algebraic Decoding of BCH Codes over Symbol-Pair Read Channels: Cases of Two-Pair and Three-Pair Error Correction

A Construction of New MDS Symbol-Pair Codes

Recently, symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One main task in symbol-pair coding theory is to design codes with large minimum pair distance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense they attain maximal minimum pair distance. In this paper, based on constacyclic codes, we construct some new MDS symbol-pair codes with minimum pair-distance five and six. Compared with classical $q$ -ary MDS codes, the constructed MDS symbol-pair codes have length up to $q^{2}+q+1$ .

A Decoding Algorithm for Cyclic Codes over Symbol-Pair Read Channels

A Decoding Algorithm for Cyclic Codes over Symbol-Pair Read Channels