Optimum Distance Profiles of Linear Block Codes

1. Optimum Distance Profiles of Linear Block Codes Yuan Luo Shanghai Jiao Tong University Xi’an Jan. 2013

2. Background: Hamming Distance Codeword with Long Length or Short Length One way is: Hamming distance, generalized Hamming Distance, … Another Direction is: Hamming distance, distance profile, … Our researches on ODP of linear block code: Golay, RS, RM, cyclic codes,…

3. Hamming Distance

4. Codeword with Long Length or Short Length Although, the linear codes with long length are most often applied in wireless communication, the codes with short length still exist in industry, for example, some storage systems, the TFCI of 3G (or 4G) system, some data with short length but need strong protection, etc.

5. For the codes with short length, the previous classic bounds can help you directly. For the codes with long length, the asymptotic forms of the previous classic bounds still work. In this topic, we consider some problems in the field of Hamming distance with short codeword length.

6. One way is: Hamming distance, generalized Hamming Distance, … Hamming distance is generalized for the description of trellis complexity of linear block codes (David Forney) and for the description of security problems (Victor Wei). We also generalized the concept to consider the relationship between a code and a subcode:

7. Another Direction is: Hamming distance, distance profile, … In the following, we consider the Hamming distance in a variational system. For example, when the encoding and decoding devices were almost selected, but the transmission rate does not need to be high in a period (in the evening not so much users), see next slide, then more redundancies can be borrowed to improve the decoding ability. What should we do to realize this idea ? And what is the principle ?

8. The TFCI in 3G system

9. Details In linear coding theory, when the number of input bits increases or decreases, some basis codewords of the generator matrix will be included or excluded, respectively.

10. For a given linear block code, we consider: ※ how to select a generator matrix and then ※ how to include or exclude the basis codewords of the generator one by one ※ while keeping the minimum distances (of the generated subcodes) as large as possible.

11. Big Problem • In general case, the algebraic structure may be lost in subcode although the properties of the original code are nice. • Then how to decode ?

12. One example • Let C be a binary [7, 4, 3] Hamming code with generator matrix G1:

13. It is easy to check that if we exclude the rows of G1 from the last to the first one by one, then the minimum distances (a distance profile) of the generated subcodes will be: • 4 4 4 • (from left to right)

14. And you can not do better, i.e. by selecting the generator matrix or deleting the rows one by one in another way, you can not get better distance profile in a dictionary order.

15. Note: we say that the sequence 3 4 6 8 is better than (or an upper bound on) the sequence 3 4 5 9 in dictionary order.

16. Another example • Let C be the binary [7, 4, 3] Hamming code with generator matrix G2:

17. It is easy to check that if we include the rows of G2 from the first to the last one by one, then the minimum distances (a distance profile) of the generated subcodes will be: 3 3 3 7 (from right to left)

18. And you can not do better, i.e. by selecting the generator matrix or adding the rows one by one in another way, you can not get better distance profile in an inverse dictionary order. • Note: we say that the sequence 3 6 8 9 is better than (or an upper bound on) the sequence 3 7 7 9 in inverse dictionary order.

19. Mathematical Description (2010 IT)

21. Optimum distance profiles

22. The Optimum Distance Profiles of the Golay Codes • For the [24, 12, 8] extended binary Golay code, we have

24. For the [23, 12, 7] binary Golay code, we have

26. For the [12, 6, 6] extended ternary Golay code, we have

27. For the [11, 6, 5] ternary Golay code, we have

28. For the researches on Reed Muller codes, see Yanling Chen’s paper (2010 IT). Maybe LDPC … in the future ?

29. To deal with the big problem, we consider cyclic code and cyclic subcode. GOOD NEWS: For general linear code, the corresponding problem is not easy since few algebraic structures are left in its subcodes. But for cyclic codes and subcodes, it looks OK

30. GOOD NEWS: For general fixed linear code, the lengths of all the distance profiles are the same as the rank of the code. For cyclic subcode chain, the lengths of the distance profiles are also the same.

31. GOOD NEWS: For general fixed linear code, the dimension profiles are the same, and any discussion is under the condition of the same dimension profile. It is unlucky that, the dimension profiles of the cyclic subcode chains are not the same, so we cannot discuss the distance profiles directly. But by classifying the set of cyclic subcode chains, we can deal with the problem.

32. Mathematics Description

33. Classification on the Cyclic Subcode Chains

34. 1 The length of its cyclic subcode chains is and J(ms) is the number of the minimal polynomials with degree ms in the factors of the generator polynomial.

35. 2 The number of its cyclic subcode chains is 3 The number of the chains in each class is:

36. 4 The number of the classes is: 5 For the special case n=qm-1, we have where is the Mobius function.

37. Example: The length of its cyclic subcode chains is 4 The number of its cyclic subcode chains is 24 The number of the chains in each class is 2 The number of the classes is 12

40. For new results about the ODP of cyclic codes, please refer to our manuscript on the punctured Reed Muller codes.

41. 谢谢