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Upper Bounds on Relative Length/Dimension Profile

Upper Bounds on Relative Length/Dimension Profile. Zhuojun Zhuang, Yuan Luo Shanghai Jiao Tong University INC. the Chinese Hong Kong University August 2012.

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Upper Bounds on Relative Length/Dimension Profile

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  1. Upper Bounds on Relative Length/Dimension Profile Zhuojun Zhuang, Yuan Luo Shanghai Jiao Tong University INC. the Chinese Hong Kong University August 2012

  2. Bounds on relative length/dimension profile (RLDP), the related bound refinement and transformation will be discussed. The results describe the security of the wiretap channel of type II and can also be applied to trellis complexity and secure network coding. RLDP is a generalization of the length/dimension profile (i.e. generalized Hamming weight) of a linear block code, one of the most famous concepts in coding theory.

  3. Agenda 1. Background 2. Upper Bounds on RLDP 3. Bound Refinement and Transformation 4. Code Constructions and Existence Bounds

  4. 1. Background The length/dimension profile (LDP) [Forney ‘94 IT], also referred as generalized Hamming weight (GHW) [Wei ‘91 IT], of a linear block code has been applied to trellis complexity (esp. in the satellite system of NASA), secure communication, multiple access communication and puncturing codes. The relative length/dimension profile (RLDP) extends LDP and has been applied to secure communication [Luo ‘05 IT], trellis complexity [Zhuang ‘11 DCC] and secure network coding [Zhang ‘09 ChinaCom/ITW].

  5. [Forney ‘94 IT] G. D. Forney, ``Dimension/length profiles and trellis complexity of linear block codes,” IEEE Trans. Inform. Theory, vol. 40, no. 6, pp. 1741-1752, 1994. [Wei ’91 IT] V. K. Wei, ``Generalized Hamming weights for linear codes,” IEEE Trans. Inform. Theory, vol. 37, no. 5, pp. 1412-1418, 1991. [Luo ’05 IT] Y. Luo, C. Mitrpant, A. J. Han Vinck, K. F. Chen, ``Some new characters on the wire-tap channel of type II,” IEEE Trans. Inform. Theory, vol. 51, no. 3, pp. 1222-1229, 2005. [Zhuang ’11 DCC] Z. Zhuang, Y. Luo, B. Dai, A. J. Han Vinck, ``On the relative profiles of a linear code and a subcode,” submitted to Des. Codes Cryptogr., under 2nd round review, 2011. [Zhang ’09 ChinaCom] Z. Zhang, ``Wiretap networks II with partial information leakage,” in 4th International Conference on Communications and Networking in China, Xi’an, China, Aug. 2009, pp. 1-5. [Zhang ’09 ITW] Z. Zhang, B. Zhuang, ``An application of the relative network generalized Hamming weight to erroneous wiretap networks,” in 2009 IEEE Information Theory Workshop, Taormina, Sicily, Italy, Oct. 2009, pp. 70-74.

  6. Wiretap Channel of Type II with Illegitimate Parties

  7. Coset Coding Scheme

  8. Security Analysis

  9. Subcode and Projection

  10. An Example

  11. Three Equivalent Concepts

  12. Three Equivalent Concepts (cont.)

  13. Bounds on Sequences

  14. Equivalence

  15. Upper Bounds on RLDP and Wiretap Channel

  16. 2. Upper Bounds on RLDP Generalized Singleton bound The bound cannot be achieved in most cases and the conditions for meeting it is rigid. Sharper bounds and code constructions?

  17. Generalized Plotkin Bound We say (C,C1) satisfying (4) meets the weak Plotkin bound.

  18. We shall see the refined generalized Plotkin bound on RLDP is always sharper than the generalized Singleton bound on RLDP.

  19. Generalized Griesmer Bound

  20. Relative Constant-Weight (RCW) Codes

  21. RCW Bound We say (C,C1) satisfying (8) meets the weak RCW bound.

  22. If C1 is a zero code, both the RCW bound and the generalized Plotkin bound on RLDP (i.e. RGHW) reduce to the generalized Plotkin bound on LDP (i.e. GHW). Otherwise, the relation is uncertain.

  23. 3. Bound Refinement and Transformation Bound Refinement

  24. Simple Refinement Without loss of generality we can always assume u is strictly increasing.

  25. Refined Bounds and Generalized Singleton Bound

  26. Bound Transformation

  27. Bound Transformation (cont.)

  28. Bound Transformation (cont.)

  29. Generalized Singleton Bounds on RDLP and IRDLP

  30. Improving Generalized Singleton Bounds

  31. An Application to Wiretap Channel

  32. 4. Code Constructions and Existence Bounds Bounds can be achieved —> Code constructions Bounds cannot be achieved —> Good code pairs —> Existence bounds Z. Zhuang, Y. Luo, B. Dai, ``Code constructions and existence bounds for relative generalized Hamming weight,” Des. Codes Cryptogr., published online, Apr. 2012.

  33. Code Constructions Indirect construction: A technique of constructing code pairs meeting a bound from the existing ones. Direct construction: Focus on the structure of generator matrices with respect to code pairs meeting bounds.

  34. Indirect Construction

  35. Code Pair Equivalence and Canonical Forms

  36. Direct Construction

  37. Code Pairs Meeting Weak Plotkin Bound

  38. Code Pairs Meeting Weak Plotkin Bound (cont.)

  39. Code Pairs Meeting Weak Plotkin Bound (cont.)

  40. Code Pairs Meeting Weak Plotkin Bound (cont.)

  41. Code Pairs Meeting Weak RCW Bound

  42. Code Pairs Meeting Weak RCW Bound (cont.)

  43. Good Code Pairs and Existence Bounds

  44. First Existence Bound

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