Binary doubly - even self - dual codes of length 72 with large automorphism groups ∗
نویسندگان
چکیده
We study binary linear codes constructed from fifty-four Hadamard 2-(71, 35, 17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss the errorcorrecting capability of the obtained codes by majority logic decoding. Further, we describe a construction of a 3-(72, 12, 11) design and a 3-(72, 16, 2010) design from a binary [72, 16, 12] code, and a construction of a strongly regular graph with parameters (126, 25, 8, 4) from a binary [35, 8, 4] code related to a derived 2-(35, 17, 16) design. AMS subject classifications: 94B05, 05B20, 05B05, 05E30
منابع مشابه
On extremal self-dual ternary codes of length 48
The notion of an extremal self-dual code has been introduced in 1 . As Gleason 2 remarks onemay use invariance properties of the weight enumerator of a self-dual code to deduce upper bounds on the minimum distance. Extremal codes are self-dual codes that achieve these bounds. The most wanted extremal code is a binary self-dual doubly even code of length 72 and minimum distance 16. One frequentl...
متن کاملOn binary self-dual extremal codes
There is a large gap between Zhang’s theoretical bound for the length n of a binary extremal self-dual doublyeven code and what we can construct. The largest n is 136. In order to find examples for larger n a non-trivial automorphism group might be helpful. In the list of known examples extended quadratic residue codes and quadratic double circulant codes have large automorphism groups. But in ...
متن کاملOn the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups
There are seven binary extremal self-dual doubly-even codes which are known to have a 2-transitive automorphism group. Using representation theoretical methods we show that there are no other such codes, except possibly n = 1 024. We also classify all extremal ternary self-dual and quaternary Hermitian self-dual codes.
متن کاملAutomorphisms of doubly-even self-dual binary codes
The automorphism group of a binary doubly-even self-dual code is always contained in the alternating group. On the other hand, given a permutation group G of degree n there exists a doubly-even self-dual G-invariant code if and only if n is a multiple of 8, every simple self-dual F2G-module occurs with even multiplicity in F n 2 , and G is contained in the alternating group.
متن کاملExtremal self-dual codes
In the present thesis we consider extremal self-dual codes. We mainly concentrate on Type II codes (binary doubly-even codes), which may theoretically exist for lengths n = 8k ≤ 3928. It is noteworthy that extremal Type II codes have been actually constructed only for 13 lengths, 136 being the largest. Over the last decades the study of extremal codes became inseparable from the study of their ...
متن کامل