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Bounds on Permutation Channel Capacity Anuran Makur Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology IEEE International Symposium on Information Theory 2020 Anuran Makur (MIT) Bounds on

  1. Bounds on Permutation Channel Capacity Anuran Makur Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology IEEE International Symposium on Information Theory 2020 Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 1 / 22

  2. Outline Introduction 1 Three Motivations Permutation Channel Model Information Capacity Achievability Bound 2 Converse Bounds 3 Conclusion 4 Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 2 / 22

  3. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  4. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  5. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  6. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  7. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Out-of-order delivery of packets Correct for packet errors/losses when packets do not have sequence numbers Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  8. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Out-of-order delivery of packets Correct for packet errors/losses when packets do not have sequence numbers Molecular/Biological Communications: [YKGR + 15], [KPM16], [HSRT17], [SH19], . . . Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  9. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Out-of-order delivery of packets Correct for packet errors/losses when packets do not have sequence numbers Molecular/Biological Communications: [YKGR + 15], [KPM16], [HSRT17], [SH19], . . . DNA based storage systems Source data encoded into DNA molecules Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  10. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Out-of-order delivery of packets Correct for packet errors/losses when packets do not have sequence numbers Molecular/Biological Communications: [YKGR + 15], [KPM16], [HSRT17], [SH19], . . . DNA based storage systems Source data encoded into DNA molecules Fragments of DNA molecules cached Receiver reads encoded data by shotgun sequencing (i.e., random sampling) Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  11. Three Motivations Coding theory: [DG01], [Mit06], [Met09], [KV15], [KT18], . . . Random deletion channel : LDPC codes nearly achieve capacity for large alphabets Codes correct for transpositions of symbols Permutation channels with insertions, deletions, substitutions, or erasures Construction and analysis of multiset codes Communication networks: [XZ02], [WWM09], [GG10], [KV13], . . . Mobile ad hoc networks, multipath routed networks, etc. Out-of-order delivery of packets Correct for packet errors/losses when packets do not have sequence numbers Molecular/Biological Communications: [YKGR + 15], [KPM16], [HSRT17], [SH19], . . . DNA based storage systems Source data encoded into DNA molecules Fragments of DNA molecules cached Receiver reads encoded data by shotgun sequencing (i.e., random sampling) Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 3 / 22

  12. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  13. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  14. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel: 2 b input symbols Alphabet symbols = all possible b -bit packets ⇒ Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  15. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel: Alphabet symbols = all possible b -bit packets Multipath routed network or evolving network topology Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  16. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel: Alphabet symbols = all possible b -bit packets Multipath routed network ⇒ packets received with transpositions Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  17. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel: Alphabet symbols = all possible b -bit packets Multipath routed network ⇒ packets received with transpositions Packets are impaired (e.g., deletions, substitutions, etc.) Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  18. Motivation: Point-to-point Communication in Packet Networks SENDER RECEIVER NETWORK Model communication network as a channel: Alphabet symbols = all possible b -bit packets Multipath routed network ⇒ packets received with transpositions Packets are impaired ⇒ model using channel probabilities Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 4 / 22

  19. Example: Coding for Random Deletion Network Consider a communication network where packets can be dropped: SENDER RECEIVER NETWORK Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 5 / 22

  20. Example: Coding for Random Deletion Network Consider a communication network where packets can be dropped: SENDER RECEIVER RANDOM RANDOM DELETION PERMUTATION NETWORK Abstraction: n -length codeword = sequence of n packets Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 5 / 22

  21. Example: Coding for Random Deletion Network Consider a communication network where packets can be dropped: SENDER RECEIVER RANDOM RANDOM DELETION PERMUTATION NETWORK Abstraction: n -length codeword = sequence of n packets Random deletion channel: Delete each symbol/packet independently with prob p ∈ (0 , 1) Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 5 / 22

  22. Example: Coding for Random Deletion Network Consider a communication network where packets can be dropped: SENDER RECEIVER RANDOM RANDOM DELETION PERMUTATION NETWORK Abstraction: n -length codeword = sequence of n packets Random deletion channel: Delete each symbol/packet independently with prob p ∈ (0 , 1) Anuran Makur (MIT) Bounds on Permutation Channel Capacity ISIT 21-26 June 2020 5 / 22

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