Decoding Frequency Permutation Arrays Under Chebyshev Distance

A frequency permutation array (FPA) of length n = mλ and distance d is a set of permutations on a multiset over m symbols, where each symbol appears exactly λ times and the distance between any two elements in the array is at least d. FPA generalizes the notion of permutation array. In this paper, under the Chebyshev distance, we first prove lower and upper bounds on the size of FPA. Then we give several constructions of FPAs, and some of them come with efficient encoding and decoding capabilities. Moreover, we show one of our designs is locally decodable, i.e., we can decode a message bit by reading at most λ+1 symbols, which has an interesting application to private information retrieval.

[1]  A.J.H. Vinck,et al.  Coded M-FSK for power line communications , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[2]  Wen-Guey Tzeng,et al.  Efficient encoding and decoding with permutation arrays , 2008, 2008 IEEE International Symposium on Information Theory.

[3]  Anxiao Jiang,et al.  Error-correcting codes for rank modulation , 2008, 2008 IEEE International Symposium on Information Theory.

[4]  Luca Trevisan,et al.  Some Applications of Coding Theory in Computational Complexity , 2004, Electron. Colloquium Comput. Complex..

[5]  Torleiv Kløve,et al.  Distance-preserving mappings from binary vectors to permutations , 2003, IEEE Trans. Inf. Theory.

[6]  Carla Golla,et al.  Flash Memories , 1999 .

[7]  Anxiao Jiang,et al.  Rank modulation for flash memories , 2008, 2008 IEEE International Symposium on Information Theory.

[8]  Moshe Schwartz,et al.  Correcting Limited-Magnitude Errors in the Rank-Modulation Scheme , 2009, IEEE Transactions on Information Theory.

[9]  A. A. Babaev.  Procedures of encoding and decoding of permutations , 1984 .

[10]  Torleiv Kløve.  Generating Functions for the Number of Permutations with Limited Displacement , 2009, Electron. J. Comb..

[11]  Sergey Yekhanin,et al.  Towards 3-query locally decodable codes of subexponential length , 2008, JACM.

[12]  H. C. Ferreira,et al.  Decoding distance-preserving permutation codes for power-line communications , 2007, AFRICON 2007.

[13]  Torleiv Kløve,et al.  Permutation Arrays Under the Chebyshev Distance , 2009, IEEE Transactions on Information Theory.

[14]  G. Mullen,et al.  Frequency permutation arrays , 2005, math/0511173.

[15]  Han Vinck,et al.  Coding and Modulation for Power-Line Communications , 2000 .

[16]  Sergey Yekhanin.  Towards 3-query locally decodable codes of subexponential length , 2007, STOC '07.

[17]  Klim Efremenko,et al.  3-Query Locally Decodable Codes of Subexponential Length , 2008 .

[18]  Kenneth W. Shum,et al.  Permutation coding and MFSK modulation for frequency selective channel , 2002, The 13th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications.

[19]  D. Slepian.  Permutation Modulation , 1965, Encyclopedia of Wireless Networks.