Recognition of a code in a noisy environment

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Basil Stevenson
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1 Introduction Recognition of a code Conclusion Université de Limoges - XLIM - DMI INRIA Rocquencourt - Projet CODES ISIT Nice - June 29, 2007

2 Introduction Recognition of a code Conclusion Outline 1 Introduction 2 Recognition of a code Problem Theoretical results Experimental results 3 Conclusion

3 Introduction Recognition of a code Conclusion Introduction Context: Data transmission. Several transformations applied to the data. Transmitted data is corrupted by noise. General Problem: Finding some information on the transformations used only from an intercepted noisy sequence.

4 Introduction Recognition of a code Conclusion Reconstruction of code Recovering the rst layer i.e. the error correcting code used. Problem of reconstruction of code. State of art: convolutional codes, turbo codes: [Fil],[Bar],[Din] linear block codes: [Val],[Clu],[Bar] This problem is NP-Hard [Val]. Complexity exponential in the error rate. Easier problem : recognition of a code.

5 Problem Let be given a binary sequence S and a binary linear code C. Is this sequence S composed of noisy codewords of C?

6 Binary linear code Let C be a binary linear code of length n and dimension k. i.e. C is a vector subspace of F n 2 of dimension k. It can be represented by: its generator matrix (whose lines span the vector subspace C ). its parity check matrix (whose lines span C ). We will use a basis of the dual code of C, H = (h 1, h 2,..., h n k ), then, for all j {1,..., n k}, for all c C, < h j, c >= n h j,i c i = 0. i=1 We will note, for x F n 2, supp(x) = {i {1,..., n} x i 0}, and w(x) = #(supp(x)) its Hamming weight.

7 Binary channel We will consider a binary symmetric channel with error rate p. S = (s j ) 1 j M is a sequence taken at the output of the channel, s j F n 2. Note: if p = 1/2, S is random (one-time pad).

8 Main idea If S comes from C : if p = 0, if 0 < p << 1/2, Else: < h i, s j >= 0 for all i,j. < h i, s j >= 1 for a few number of i,j. < h i, s j >= 1 half of the time.

9 Dierent cases Let h be a non-zero word of C. If S comes from a random sequence: dim(h #(h ) = n 1, ) #(F n 2 ) = 1 2, P [< h, s j >= 1] = 1/2. If S comes from words of C : For all j, s j = c j + e j, with c j C and e j the error vector. < h, s j >= 1 #(supp(h) supp(e j )) is odd. P [< h, s j >= 1] = 1 (1 2p)w(h) 2.

10 Dierent cases If S comes from a sequence of words of C C : There exists at least one h 0 in a basis of C such that h 0 C, for this h 0, P [< h 0, s j >= 1] = 1/2. Other cases: heuristic result but conrmed by experiments. P [< h, s j >= 1] = 1/2.

11 Statistical test Summary: If S does not come from C : There exists at least one h {h 1,..., h n k }, s.t. If S comes from C : For all h {h 1,..., h n k }, P [< h, s j >= 1] = 1/2. P [< h, s j >= 1] = (1 2p)w(h). Idea: M Is < h, s j > (sum in Z) close to M 2 M 2 (1 2p)w(h)? j=1

12 Theorem Theorem: The statistical test consisting in deciding that S = (s j ) 1 j M comes from a sequence of M words of h if and M only if < h, s j > T with j=1 M = ( ) 2 b 1 (1 2p) 2w(h) a, (1 2p) w(h) T = 1 2 (M + a M) and a = φ 1 (α), b = φ 1 (1 β) veries: M P[( < h, s j >) T S is random] = α (false alarm), j=1 M P[( < h, s j >) T S comes from h ] = β (non detection). j=1

13 Input: C a [n, k]-linear binary code, a binary symmetric channel with error rate p, S = (s j ) 1 j M a sequence taken at the output of the channel, α,β false alarm and non detection probabilities. Initialization: Compute (h 1,..., h n k ) a basis of C with low weight words. Compute M i and T i for each i {1,..., n k}. Algorithm: M i N i = < h i, s j > (sum in Z) for i {1,..., n k}. Output: j=1 If N i T i for all i {1,..., n k}, say that S comes from a sequence of words of C. If N i T i for at least one i {1,..., n k}, say that S does not come from a sequence of words of C.

14 Computing results Code w(h) Error Time used n max rate p M T is s BCH BCH RM RM Random Random Here, α = β = 10 6.

15 Synchronization Input: C, p, S, α and β. Initialization: Compute S (l) = (s (l) j ) 1 j M, 0 l n 1. Compute (h 1,..., h r ) some words of a basis of C. Compute M i and T i for each i {1,..., r}. Algorithm: For l from 0 to n 1 do M N (l) i i = < h i, s (l) j > (sum in Z) for i {1,..., r}. j=1 Output: If N (l) i T i for all i {1,..., r}, say that S seems to come from a sequence of words of C with synchronization l check it!

16 Introduction Recognition of a code Conclusion Conclusion Easy and fast algorithm to recognize a code. Can reach high error rate (compared to reconstruction). Application to synchronization.

17 Introduction Recognition of a code Conclusion Thanks for your attention.

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