Order-Optimal Permutation Codes in the Generalized Cayley Metric - PowerPoint PPT Presentation

Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek LORIS, Electrical and Computer Engineering, UCLA March 12th, 2018 Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation Codes in the Generalized Cayley Metric NVMW2018 1 / 20

Outline Decoding Schemes NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek Conclusion and Future Work Conclusion 4 Systematic Codes Rate Analysis Encoding Schemes 1 Construction 3 Order-Optimal Codes Distances of Interest Theoretical Analysis 2 Objective Background Motivation 2 / 20

Motivation Encoding Schemes NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek Conclusion and Future Work Conclusion 4 Systematic Codes Rate Analysis Decoding Schemes Construction Outline 3 Order-Optimal Codes Distances of Interest Theoretical Analysis 2 Objective Background Motivation 1 3 / 20

Motivation Background Applications Flash memories: charge leakage between cells [1] Genome resequencing: gene rearrangement in a chromosome Cloud storage system: rearrangements of items in multiple folders [1] A. Jiang et al. “Rank Modulation for Flash Memories”. In: IEEE Trans. Inf. Theory 55.6 (2009), pp. 2659–2673. Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation Codes in the Generalized Cayley Metric NVMW2018 4 / 20

Motivation Background Applications Flash memories: charge leakage between cells [1] Genome resequencing: gene rearrangement in a chromosome [2] Cloud storage system: rearrangements of items in multiple folders [1] A. Jiang et al. “Rank Modulation for Flash Memories”. In: IEEE Trans. Inf. Theory 55.6 (2009), pp. 2659–2673. In: Journal of Computational Biology 24 (2017), pp. 127–137. Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation Codes in the Generalized Cayley Metric NVMW2018 4 / 20 [2] R. Zeira and R. Shamir. “Sorting by cuts, joins and whole chromosome duplications”.

Motivation Objective Measures in Permutation Codes Common measures Kendall- metric: transpositions Ulam metric: translocation Measure under discussion Generalized Cayley metric: generalized transposition No restrictions on the lengths and positions of the translocated segments Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation Codes in the Generalized Cayley Metric NVMW2018 5 / 20

Motivation Objective Measures in Permutation Codes Common measures Ulam metric: translocation Measure under discussion Generalized Cayley metric: generalized transposition No restrictions on the lengths and positions of the translocated segments In: IEEE Trans. Inf. Theory 62 (Jan. 2016), pp. 151–158. Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation Codes in the Generalized Cayley Metric NVMW2018 5 / 20 Kendall- τ metric: transpositions [3] [3] Y. Zhang and G. Ge. “Snake-in-the-Box Codes for Rank Modulation Under Kendall’s τ -Metric”.

Motivation In: IEEE Trans. Inf. Theory 62 (Jan. 2016), NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek (May 2013), pp. 3003–3020. In: IEEE Trans. Inf. Theory 59 pp. 151–158. 5 / 20 Objective No restrictions on the lengths and positions of the translocated segments Generalized Cayley metric: generalized transposition Measure under discussion Ulam metric: translocation [4] Common measures Measures in Permutation Codes Kendall- τ metric: transpositions [3] [3] Y. Zhang and G. Ge. “Snake-in-the-Box Codes for Rank Modulation Under Kendall’s τ -Metric”. [4] F. Farnoud, V. Skachek, and O. Milenkovic. “Error-correction in Flash Memories via Codes in the Ulam Metric”.

Motivation Objective NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek Inf. Theory . Hawaii, USA, June 2014, pp. 2959–2963. In: Proc. IEEE Int. Symp. (May 2013), pp. 3003–3020. In: IEEE Trans. Inf. Theory 59 pp. 151–158. In: IEEE Trans. Inf. Theory 62 (Jan. 2016), No restrictions on the lengths and positions of the translocated segments Generalized Cayley metric: generalized transposition [5] Measure under discussion Ulam metric: translocation [4] Common measures Measures in Permutation Codes 5 / 20 Kendall- τ metric: transpositions [3] [3] Y. Zhang and G. Ge. “Snake-in-the-Box Codes for Rank Modulation Under Kendall’s τ -Metric”. [4] F. Farnoud, V. Skachek, and O. Milenkovic. “Error-correction in Flash Memories via Codes in the Ulam Metric”. [5] Y. M. Chee and V. K. Vu. “Breakpoint analysis and permutation codes in generalized Kendall tau and Cayley metrics”.

Motivation Ultimate goal NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek t log N bits errors: Redundancy for an order-optimal code that corrects t generalized transposition codelength Objective N bits, where N is the Interleaving based: induce a redundancy of Based on the error-correcting codes in the Ulam metric Prior work Construction of order-optimal codes in the generalized Cayley metric Objective Ultimate Goal 6 / 20

Motivation Redundancy for an order-optimal code that corrects t generalized transposition NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek Inf. Theory . Hawaii, USA, June 2014, pp. 2959–2963. In: Proc. IEEE Int. Symp. [6] Y. M. Chee and V. K. Vu. “Breakpoint analysis and permutation codes in generalized Kendall tau and Cayley metrics”. t log N bits errors: Ultimate goal Objective codelength N bits, where N is the Interleaving based: induce a redundancy of Based on the error-correcting codes in the Ulam metric Prior work [6] Construction of order-optimal codes in the generalized Cayley metric Objective Ultimate Goal 6 / 20

Motivation Objective NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek (May 2013), pp. 3003–3020. In: IEEE Trans. Inf. Theory 59 Inf. Theory . Hawaii, USA, June 2014, pp. 2959–2963. In: Proc. IEEE Int. Symp. [6] Y. M. Chee and V. K. Vu. “Breakpoint analysis and permutation codes in generalized Kendall tau and Cayley metrics”. t log N bits errors: Redundancy for an order-optimal code that corrects t generalized transposition Ultimate goal codelength N bits, where N is the Interleaving based: induce a redundancy of Based on the error-correcting codes in the Ulam metric [7] Prior work [6] Construction of order-optimal codes in the generalized Cayley metric Objective Ultimate Goal 6 / 20 [7] F. Farnoud, V. Skachek, and O. Milenkovic. “Error-correction in Flash Memories via Codes in the Ulam Metric”.

Motivation t log N bits NVMW2018 Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek (May 2013), pp. 3003–3020. In: IEEE Trans. Inf. Theory 59 Inf. Theory . Hawaii, USA, June 2014, pp. 2959–2963. In: Proc. IEEE Int. Symp. [6] Y. M. Chee and V. K. Vu. “Breakpoint analysis and permutation codes in generalized Kendall tau and Cayley metrics”. errors: Objective Redundancy for an order-optimal code that corrects t generalized transposition Ultimate goal codelength Based on the error-correcting codes in the Ulam metric [7] Prior work [6] Construction of order-optimal codes in the generalized Cayley metric Objective Ultimate Goal 6 / 20 Interleaving based: induce a redundancy of O ( N ) bits, where N is the [7] F. Farnoud, V. Skachek, and O. Milenkovic. “Error-correction in Flash Memories via Codes in the Ulam Metric”.

Order-Optimal Permutation Codes in the Generalized Cayley Metric - PowerPoint PPT Presentation

Order-Optimal Permutation Codes in the Generalized Cayley Metric Siyi Yang , Clayton Schoeny, Lara Dolecek LORIS, Electrical and Computer Engineering, UCLA March 12th, 2018 Siyi Yang , Clayton Schoeny, Lara Dolecek Order-Optimal Permutation

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