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Consolidation for compact constraints and Kendall tau LP decodable permutation codes

Abstract

Invented in the 1960s, permutation codes have reemerged in recent years as a topic of great interest because of properties making them attractive for certain modern technological applications, especially flash memory. In 2011 a polynomial time algorithm called linear programming (LP) decoding was introduced for a class of permutation codes where the feasible set of codewords was a subset of the vertex set of a code polytope. In this paper we investigate a new class of linear constraints for matrix polytopes with no fractional vertices through a new concept called “consolidation.” We then introduce a necessary and sufficient condition for which LP decoding methods, originally designed for the Euclidean metric, may be extended to provide an efficient decoding algorithm for permutation codes with the Kendall tau metric.

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Notes

  1. 1.

    In some references, G is chosen as a generalized permutation group, e.g., a signed permutation group.

  2. 2.

    The original problem is to maximize \(\mathrm {Trace}( \varvec{\mu } \varvec{\lambda }^T X )\). It is directly obtained that \( \mathrm {Trace}( \varvec{\mu } \varvec{\lambda }^T X ) = \mathrm {Trace}( \varvec{\lambda }^T X \varvec{\mu } ) = \varvec{\lambda }^T X \varvec{\mu }\).

  3. 3.

    Televis: a toy consisting of two balls and a string which connects the balls.

  4. 4.

    In some references, a dihedral group of degree n is denoted by \(D_n\).

  5. 5.

    This notation is defined in Sect. 5.1, Definition 15.

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Acknowledgements

The authors would like to thank Dr. JB Nation for his help proofreading. We are also grateful to the anonymous reviewers of this journal for many valuable insights that have significantly improved the quality of this paper. This paper is partially supported by KAKENHI(B) 16K12391, 16K06336, and 26289116.

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Correspondence to Justin Kong.

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Communicated by P. Charpin.

Appendix

Appendix

Definition 22

(Linear constraints, satisfy, \(\models \)) A linear constraint l(X) for an n-by-n matrix is defined as either a linear equation or linear inequality on entries of a matrix.

Formally speaking, by regarding an entry \(X_{i,j}\) as a variable (\(0 \le i,j < n\)), we state either

$$\begin{aligned} l(X) : \sum _{0 \le i,j < n} c_{i,j} X_{i,j} = c_0, \end{aligned}$$

or

$$\begin{aligned} l(X) : \sum _{0 \le i,j < n} c_{i,j} X_{i,j} \ge c_0, \end{aligned}$$

for some \(c_0, c_{i,j} \in {\mathbb {R}}\). The relation \(=\) or \(\ge \) is uniquely determined by l(X). Instead of the symbols \(=\) and \(\ge \), we may use \(\trianglerighteq _l\) (or simply \(\trianglerighteq \)), e.g.,

$$\begin{aligned} l(X) : \sum _{0 \le i,j < n} c_{i,j} X_{i,j} \trianglerighteq _l c_0. \end{aligned}$$

If we do not need to clarify the variable X of a linear constant l(X), we denote it simply by l.

For a linear constraint \(l \in {\mathcal {L}}\) and a matrix \(X \in \mathrm {M}_n ({\mathbb {R}})\), if X satisfies l, we write \(X \models l\). If \(X \models l\) for every \(l \in {\mathcal {L}}\), we write \(X \models {\mathcal {L}}\).

Definition 23

(Birkhoff polytope, doubly stochastic polytope) Let \({\mathcal {L}} \) be a doubly stochastic constraint for an n-by-n matrix. The collection of n-by-n matrices which satisfy all linear constraints in \({\mathcal {L}}\) is denoted by \({\mathcal {D}}_n [{\mathcal {L}}]\). We call \({\mathcal {D}}_n [ {\mathcal {L}}]\) a doubly stochastic polytope of \({\mathcal {L}}\).

For a subset \({\mathcal {D}} \subset \mathrm {M}_n ( {\mathbb {R}} )\), \({\mathcal {D}}\) is said to be a doubly stochastic polytope if there exists a doubly stochastic constraint \({\mathcal {L}}\) such that \({\mathcal {D}} = {\mathcal {D}}_n [ {\mathcal {L}} ]\). The polytope \(\mathrm {DSM}_n\) is an example of doubly stochastic polytope. \(\mathrm {DSM}_n\) is said to be a Birkhoff polytope.

Definition 24

(Vertex) Let \({\mathcal {D}}\) be a doubly stochastic polytope.

An element \(X \in {\mathcal {D}}\) is said to be a vertex if there are neither elements \(X_0, X_1 \in {\mathcal {D}}\) with \(X_0 \ne X_1\) nor positive numbers \(c_0, c_1 \in {\mathbb {R}},\) \(c_0 + c_1 = 1\), such that

$$\begin{aligned} X = c_0 X_0 + c_1 X_1. \end{aligned}$$

We denote the set of vertices for \({\mathcal {D}}\) by \(\mathrm {Ver}( {\mathcal {D}} )\).

Definition 25

(Kendall tau distance) Given \(\sigma ,\tau \in S_n\), the Kendall tau distance \(\mathrm {d}_K(\sigma ,\tau )\) between \(\sigma \) and \(\tau \) is defined as

$$\begin{aligned} \mathrm {d}_K(\sigma ,\tau ):= \left| \left\{ (i,j) \;\vert \; 0\le i < j \le n-1, \big (\sigma ^{-1}\tau \big )_i > \big (\sigma ^{-1}\tau \big )_j\right\} \right| , \end{aligned}$$

where |A| denotes the cardinality of the set A.

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Hagiwara, M., Kong, J. Consolidation for compact constraints and Kendall tau LP decodable permutation codes. Des. Codes Cryptogr. 85, 483–521 (2017). https://doi.org/10.1007/s10623-016-0313-5

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Keywords

  • Coding and information theory
  • Graphs and abstract algebra
  • Linear programming
  • Permutation groups
  • Decoding

Mathematics Subject Classification

  • 68P30
  • 05C25
  • 90C05
  • 20B
  • 94B35