Extremal Self-Dual Binary Codes With Automorphisms Of Type 3-(8, 14)

With the development of mathematical theory and IT technology, besides binary codes, now codes over other finite fields and even over finite rings are also studied extensively. So for linear codes , we have now a more general form:Definition Let Fq be a field with q elements and let Fq n be the n dimentional row vector space over Fq. A subspace of Fq n is called a linear q-ary code of length n. If the code C has dimension k , then C is called an [ n, k] code over Fq .Though the binary codes were the main object in the past , today it also attracts lots of people's attention. Especially, lots of important results were found about its self-dual codes.Definition For x∈F2n, the number wt (x)of components of x equal to 1 is called the Hamming weight of x .Definition For x = ( x1 , , xn ), y = ( y1 , , y n)∈F2n, define called the Euclidean inner product of x, y.Definition The dual code of C is the code The code C is self-dual if C = C⊥.Clearly, if C is an [ n, k] code, C⊥is an [ n , n?k] code. Furthermore if C is self-dual, then n must be even.Definition We call a binary code to be even , if its each codeword has an even weight.Definition A self-dual code with all codewords of weight divisible by 4 is called doubly-even or Type II; a self-dual code with some codeword of weight not divisible by 4 is called Type I.Definition The minimal weight of C is the minimum nonzero weight of any codeword in C . If C is an [ n, k] code with minimal weight d , then it is called an [ n , k,d] code.The minimal weight d of a linear code measure the capability correcting errors for the code. For a certain code length n , a code with a larger minimal weight will be better for us. This leads to lots of studies on the upper bounds on the minimum weight of self-dual codes.The first was given by C. Mallows and N.J.A Sloane in 1973.Theorem 1 Let C be an [ n, n2,d] self-dual binary code. Then (i) d≤2 ?n 8? +2; (ii) d≤4 ?n 24? +4, whenever C is Type II.Then an improvement was made by Rains in 1998.Theorem 2 Let C be an [ n, n2,d] self-dual binary code. (i) If n≠22(mod24), then d≤4 ?n 24? +4; (ii) If n≡22(mod24), then d≤4 ?n 24? +6.Definition Type I codes meeting the bound of Theorem 1 and Type II codes meeting the bound of Theorem 2(ii) are called extremal.Definition Two codes are equivalent if there exists a permutation matrix M such that C1 M= C2. A permutation matrix is a matrix that each rows and column has exactly one 1, and the others are 0s.People try to classify binary self-dual codes up to equivalence. So far all binary self-dual codes have been classified for length n with n≤32. But there will be a large amount of codes and it is virtually impossible to classify it completely when n≥34. So people mainly studied the extremal codes for codes with larger code length.It is difficult to classify general extremal codes. So some restriction is put on the extremal code to classify it.Let p be an odd prime, C be an [n,k] linear code over F2 . Then all the permutationsσsuch thatσC =C form a group, called the automorphism group of C, and elements of C are called the automorphisms of C . If an automorphismσhas c p-cycles and f = n?cp fixed points , thenσis called a automorphism of typeFor n=38, it was showed that there exist only four possible types for automorphisms of order 3: 3-(6,20), 3-(8,14), 3-(10,8) and 3-(12,2). But the existence of corresponing extremal codes for each of them were unknown except for the last case. Based on reults of Huffman, Song Wentu showed in his master's thesis in 2006 that a linear binary code C is a self-dual code with an automorphism of type 3-(c,and only if C=D⊕E, where D is a self-dual binary code, E is a Hermite self-dual code over F4 . Fan Jiqiu found selfdual codes with automorphisms of type 3-(12,2) by presenting generator matrix in his master's thesis in 2007.Similar to the minimal weight , the weight enumerator of a code C is also very important for correcting. The weight enumerator of C is: where Ai is the numbers of code words in C with weight i .There exist two possible weight enumerator for extremal selfdual codes of length n=38. In this thesis we find extremal selfdual codes with automorphisms of type 3-(8,14). By the a result of Song Wentu, we need to find the generator matrix of self-dual binary codes of length 22 and the those of Hermite self-dual codes over F4 of length 8. Pless in 1975 and MacWilliams in 1978 had given what we need. There are 25 non-equivalent generator matrixes, 17 decomposable and 8 indecomposable.In fact we just need to consider the 8 indecomposable ones due to the following theorems:Theorem 3.2 Let M' be a generator matrix of a self-dual binary code of length 22 and d be the minimal weigth of M'. If M' can be a part of G3 8, then d≥4.Theorem 3.3 Let M' be a generator matrix of a self-dual binary code of length 22. If M' is decomposable, then M' can not be a part of G3 8.Among the 8 indecomposable generator matrixes we just need to consider the G 22. By a theorem of Song Wentu, we need to find out the possible permutations among all permutations (21! in all), and 10 can be found by performing following algorithm.Algorithm 1:①take 8 columns of S in certain order;②take the 77×8 matrix;③if there is a row in which the number of 1 smaller than 3, then do①; else output the orders corresponding to the 8 colomns and then do①. By a theorem of Song Wentu, using the generator matrix of Hermite self-dual code over F4 of length 8 and the corresponding matrices to the above permutations, we finally find the desired generator matrix by doing the following algorithm:Algorithm 2:①take a permutation E8 ' of E8 in certain order;②expand E8 ' to H ;③set④test the extremenesst of G⑤if there is a codeword of G whose weigth is smaller than 8, then do①, else output G , then do①The generator matrix of the extremal self-dual binary code with automorphism of type 3-(8,14) we find is as follows: Using a simple examination we find that G3 8 meets weight enumerator W2 .When runing the procedure we also obtained lots of generator matrices. They also were the generator matrices of extremal self-dual codes with an automorphism of type 3-(8,14). However we failed to classify them up to equivalence.