1. Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes Kenji Yasunaga * Toru Fujiwara + * Kwansei Gakuin University, Japan + Osaka University, Japan 2008 IEEE International Symposium on Information Theory, July 6th to 11th, 2008 Sheraton Centre Toronto Hotel, Toronto, Ontario, Canada

2. 2 Summary of the Work Main Result A lower bound on #(uncorrectable errors of weight ) for binary linear codes. d : the minimum distance of the code A generalization to weight. Main Techniques Monotone error structure (Larger half) Monotone error structure appears in [Peterson, Weldon, 1972]. Larger half was introduced in [Helleseth, Kløve, Levenshtein, 2005].

3. 3 Outline Correctable/Uncorrectable Errors Our Results Monotone Error Structure Proof Sketch of Our Results

4. 4 Outline Correctable/Uncorrectable Errors Our Results Monotone Error Structure Proof Sketch of Our Results

5. 5 Problem Setting Binary linear code C ⊆ {0,1} n Error vector e ∊ {0,1} n If w(e) < d/2 ⇒ e is always correctable. If w(e) ≥ d/2 ⇒ ? w(x) : the Hamming weight of x In this work, we investigate #( correctable errors of weight i ) for i ≥ d/2.

6. 6 Correctable/Uncorrectable Errors Correctable errors E 0 (C) = Correctable by minimum distance decoding. E i 0 (C) : Correctable errors of weight i Uncorrectable errors E 1 (C) = {0,1} n 〵 E 0 (C) E i 1 (C) : Uncorrectable errors of weight i The error probability over BSC p is. Minimum distance decoding Outputs a nearest (w.r.t. Hamming dist.) codeword to the input. Performs ML decoding for BSC. Syndrome decoding is a minimum distance decoding.

7. 7 Syndrome Decoding Coset partitioning Syndrome decoding Output y + v i if y ∈ C i ( y is the input). Coset leaders = Correctable errors. : Coset of C : Coset leader of C i

8. 8 Outline Correctable/Uncorrectable Errors Our Results Monotone Error Structure Proof Sketch of Our Results

9. 9 Previous Results for |E i 1 (C)| For the first-order Reed-Muller code RM m |E d/2 1 (RM m )| [Wu, 1998] |E d/2+1 1 (RM m )| [Yasunaga, Fujiwara, 2007] For binary linear codes Upper bounds on |E i 1 (C)| for every 0 ≤ i ≤ n [Poltyrev 1994], [Helleseth, Kløve 1997], [Helleseth, Kløve, Levenshtein 2005]

10. 10 Our Results A lower bound on for codes satisfying some condition. The condition is not too restrictive.  Long Reed-Muller codes and random linear codes satisfy Given by #(codewords of weight d (and d+1 )). Asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. A generalization to for. The bound is weak.

11. 11 Outline Correctable/Uncorrectable Errors Our Results Monotone Error Structure Proof Sketch of Our Results

12. 12 Monotone Error Structure Recall that a coset leader is a minimum weight vector in a coset. There may be more than one minimum weight vector in the same coset. ⇒ Any of them will do. If we take the lexicographically smallest one for all cosets, ⇒ Correctable/uncorrectable errors have a monotone structure.

13. 13 Monotone Error Structure Notation Support of v : S(v) = { i : v i ≠ 0 } v is covered by u : S(v) ⊆ S(u) Monotone error structure v is correctable. ⇒ All vectors that are covered by v are correctable. v is uncorrectable. ⇒ All vectors that cover v are uncorrectable. Example 1100 is correctable. ⇒ 0000, 1000, 0100 are correctable. 0011 is uncorrectable. ⇒ 1011, 0111, 1111 are uncorrectable.

14. 14 Minimal Uncorrectable Errors Errors have the monotone structure (w.r.t ⊆ ). ⇒ E 1 (C) is characterized by minimal vectors (w.r.t. ⊆ ). Minimal uncorrectable errors M 1 (C) = Uncorrectable errs. that are not covered by other uncorrectable errs. M 1 (C) uniquely determines E 1 (C). Larger half LH(c) of c ∈ C Introduced for characterizing M 1 (C) in [Helleseth et al., 2005]. Combinatorial construction is given in [Helleseth et al., 2005]. M 1 (C) ⊆ LH(C 〵 {0}) ⊆ E 1 (C), where.

15. 15 Outline Correctable/Uncorrectable Errors Previous Results Our Results Monotone Error Structure Proof Sketch of Our Results

16. 16 Proof Sketch of Our Results Objective : To derive a lower bound on. The following equalities hold: [Proof] Since is the smallest weight in E 1 (C), uncorrectable errors of weight do not cover any other uncorrectable errors. ⇒ Derive a lower bound on.

17. Proof Sketch of Our Results ( d is even ), where A i (C) = { codewords of weight i in C}. Larger halves of two codewords in A d (C) are almost disjoint. for every 17 = = LH( ・ ) c1c1 c2c2 c4c4 c3c3 Ad(C)Ad(C) ≤ |A d (C)| – 1 |LH(c i )| For every c i ∈ A d (C), #(common LH) is less than |A d (C)|.

18. The Results ( d is even ) When d is even, if holds, then If as then upper and lower bounds asymptotically coincide.  For Reed-Muller codes and random linear codes, the upper and lower bounds asymptotically coincide. 18 Upper bound is from [Helleseth et al. 2005]

19. The Results ( d is odd ) 19 When d is odd, if holds, then If as then upper and lower bounds asymptotically coincide. Upper bound is from [Helleseth et al. 2005]

20. A similar argument can be applied to weight. For large i The condition for the bound is more restrictive. The bound is weak. The bound is a lower bound on LH i (C). The difference between LH i (C) and E i 1 (C) is large. A Generalization to Larger Weights 20 For an integer i with ⌈ d/2 ⌉ ≤ i ≤ ⌊ n/2 ⌋, if holds, then where B i = |A 2i−2 (C)| +|A 2i−1 (C)| + |A 2i (C)|.

21. Conclusion Main results A lower bound on #(correctable errors of weight ) for binary linear codes satisfying some condition. The bound asymptotically coincides with the upper bound for Reed- Muller codes and random linear codes. Monotone error structure & larger half are main tools. A generalization to weight is also obtained.  The generalized bound is weak for large i. Future work A good lower bound for weight. 21

22. Codes Satisfying the Condition The condition Codes satisfying the condition (n, k) primitive BCH codes for n = 127 and k ≤ 64, n = 63 and k ≤ 24 (n, k) extended primitive BCH codes for n = 127 and k ≤ 64, n = 63 and k ≤ 24 r -th order Reed-Muller codes of length 2 m fixed r and m →∞ Random linear codes for n → ∞ 22 rm 1≥ 4 2≥ 6 3≥ 8 4≥ 10 5≥ 11 6≥ 13 for even d for odd d

23. 23 Proof Sketch of Our Results ( d is even ) x2x2 x1x1

24. 24 Proof Sketch of Our Results ( d is even ) c1c1 c2c2 c4c4 c3c3 Ad(C)Ad(C) A i (C) : the set of codewords with weight i x1x1 x2x2

25. 25 Proof Sketch of Our Results ( d is even ) LH( ・ ) x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 Ad(C)Ad(C) A i (C) : the set of codewords with weight i

26. 26 Proof Sketch of Our Results ( d is even ) x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

27. 27 Proof Sketch of Our Results ( d is even ) x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

28. 28 Proof Sketch of Our Results ( d is even ) x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

29. 29 Proof Sketch of Our Results ( d is even ) |LH(c i )| x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

30. 30 Proof Sketch of Our Results ( d is even ) |LH(c i )| ≤ |A d (T)| – 1 x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

31. 31 Proof Sketch of Our Results ( d is even ) For every c i ∈ A d (T), #(common LH) is less than |A d (T)| – 1 Thus, if |LH(c i )| > |A d (T)| – 1, then (|LH(c i )| － |A d (T)|+1) |A d (T)| ≤ |LH d/2 (T)| |LH(c i )| ≤ |A d (T)| – 1 x1x1 x2x2 c1c1 c2c2 c4c4 c3c3 LH( ・ ) Ad(C)Ad(C) A i (C) : the set of codewords with weight i

32. A Generalization to Larger Weight The lower bound for weight is obtained by considering the vectors of weight in A similar argument can be applied to weight However, for large i, The condition for the bound is more restrictive The bound is weak  The bound is a lower bound on LH i (C)  The difference between LH i (C) and E i 1 (C) is large 32