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1、1yuan luoxian jan. 2013optimum distance profiles of linear block codesshanghai jiao tong universityhamming distancecodeword with long length or short lengthone way is: hamming distance, generalized hamming distance, another direction is: hamming distance, distance profile, background:our researches

2、on odp of linear block code:golay, rs, rm, cyclic codes,3hamming distance4although, 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 wit

3、h short length but need strong protection, etc. codeword with long length or short length5for 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

4、problems in the field of hamming distance with short codeword length.6hamming 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

5、between a code and a subcode:one way is: hamming distance, generalized hamming distance, 7in 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 peri

6、od (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 ?another direction is: hamming distance, distance profile, 8the tfci in 3g system9detailsin linear coding t

7、heory, 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

8、 one by one while keeping the minimum distances (of the generated subcodes) as large as possible.big problemin general case, the algebraic structure may be lost in subcode although the properties of the original code are nice.then how to decode ?1112one examplelet c be a binary 7, 4, 3 hamming code

9、with generator matrix g1:13l 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:34 4 4 (from left to right)14and you can not do better, i.e. by selecting the generator matrix o

10、r deleting the rows one by one in another way, you can not get better distance profile in a dictionary order.15note: we say that the sequence 3 4 6 8is better than (or an upper bound on) the sequence3 4 5 9in dictionary order.16another examplelet c be the binary 7, 4, 3 hamming code with generator m

11、atrix g2:17it 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)18and you can not do better, i.e. by selecting the generator matrix or adding the row

12、s 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 9is better than (or an upper bound on) the sequence3 7 7 9in inverse dictionary order.19mathematical description (2010 it)2021optimum distance profiles22the optimu

13、m distance profiles of the golay codesfor the 24, 12, 8 extended binary golay code, we have2324for the 23, 12, 7 binary golay code, we have2526l for the 12, 6, 6 extended ternary golay code, we have27l for the 11, 6, 5 ternary golay code, we have28for the researches on reed muller codes, see yanling

14、 chens paper (2010 it).maybe ldpc in the future ?29to 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

15、30good 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.31good news:for general fixed linear code, the dimension profiles are the same, and any discussio

16、n 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.32mathematics descr

17、iption33classification on the cyclic subcode chains 341 the length of its cyclic subcode chains ismmmssssmjml|:)()(and j(ms) is the number of the minimal polynomials with degree ms in the factors of the generator polynomial.35!)()(|:mmmssssmjml2 the number of its cyclic subcode chains is3 the number of the chains in each class is:!)()(|:mmmssssmjml364 the number of the classes is:5 for the special case n=qm-1, we have1)(1)(|/smrrmssmforqrmmlssmmmssmmmssssssmjmlmjml|:|:!)()(!)()(where is the mobius function.)(37example:th