Upper Bounds on Relative Length/Dimension ProfileUpper Bounds on Relative Length/Dimension Profile

Zhuojun Zhuang, Yuan Luo

Shanghai Jiao Tong University

INC. the Chinese Hong Kong UniversityAugust 2012

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.

1. Background

2. Upper Bounds on RLDP

3. Bound Refinement and Transformation

Agenda

4. Code Constructions and Existence Bounds

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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].

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[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.

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Wiretap Channel of Type II with Illegitimate Parties

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Coset Coding Scheme

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Security Analysis

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Subcode and Projection

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An Example

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Three Equivalent Concepts

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Three Equivalent Concepts (cont.)

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Bounds on Sequences

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Equivalence

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Upper Bounds on RLDP and Wiretap Channel

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2. Upper Bounds on RLDP

The bound cannot be achieved in most cases and the conditions for meeting it is rigid.Sharper bounds and code constructions?

Generalized Singleton bound

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Generalized Plotkin Bound

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

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

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Generalized Griesmer Bound

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Relative Constant-Weight (RCW) Codes

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RCW Bound

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

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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.

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3. Bound Refinement and Transformation

Bound Refinement

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Simple Refinement

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

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Refined Bounds and Generalized Singleton

Bound

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Bound Transformation

Bound Transformation (cont.)

Bound Transformation (cont.)

Generalized Singleton Bounds on RDLP and IRDLP

Improving Generalized Singleton Bounds

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An Application to Wiretap Channel

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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.

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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.

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Indirect Construction

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Code Pair Equivalence and

Canonical Forms

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Direct Construction

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Code Pairs Meeting Weak Plotkin Bound

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Code Pairs Meeting Weak Plotkin Bound (cont.)

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Code Pairs Meeting Weak Plotkin Bound (cont.)

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Code Pairs Meeting Weak Plotkin Bound (cont.)

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Code Pairs Meeting Weak RCW Bound

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Code Pairs Meeting Weak RCW Bound (cont.)

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Good Code Pairs and Existence Bounds

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First Existence Bound

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Asymptotic Gilbert-Varshamov Bound

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Second Existence Bound

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Lower Bound on RLDP

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Asymptotic Equivalence of Existence Bounds

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Asymptotic Equivalence of Existence Bounds

(cont.)

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Validity of Existence Bounds