New Uniqueness Proofs for the (5, 8, 24), (5, 6, 12) and Related Steiner Systems
New Uniqueness Proofs for the (5, 8, 24), (5, 6, 12) and Related Steiner Systems
复制标题
(5,8,24),(5,6,12)和相关斯坦纳系统的新唯一性证明
DOI：
10.1016/0097-3165(82)90039-5
复制
发表时间：
1982
期刊：
影响因子：
--
通讯作者：
Deborah J. Bergstrand
中科院分区：
文献类型：
--
作者：
Deborah J. Bergstrand
利用编码理论给出了某些Steiner系统唯一性的新的初等证明。在这个过程中，一些涉及的代码被证明是唯一的。（5，8，24）Steiner系统的唯一性证明是由于约翰康威。系统的块用于生成长度为24的二进制码。任何两个这样的代码，然后被证明是等价的，直到一个置换的坐标。在（4，7，23）系统的唯一性证明中，块生成的长度为23的码被扩展为长度为24的码。这个较大代码的最小权重向量保持（5，8，24）Steiner系统。这个结果和前一个结果一起完成了证明。继续（3，6，22）Steiner系统的唯一性结果，块产生长度为22的码，通过添加两个坐标和一个附加矢量，该码被扩展为相同长度的24码。这种扩展最终需要计算长度为22的码的陪集权重分布，这是迄今为止未知的结果。本文用CAMAC计算机系统计算了（22，11，6）自对偶码的完全陪集重量分布，对（5，6，12）和（4，5，11）Steiner系统作了不同的处理。它示出，每个系统是完全由6块的选择，可以假设躺在任何这样的设计确定。这六个块实际上形成对应于两个系统的长度为12（和11）的三进制码的基础，并且可以由独立于设计的算法生成。给出了该算法，并利用CAMAC系统计算了所得码的最小权向量，即最佳三元Golay码及其扩展码。
New elementary proofs of the uniqueness of certain Steiner systems using coding theory are presented. In the process some of the codes involved are shown to be unique.The uniqueness proof for the (5, 8, 24) Steiner system is due to John Conway. The blocks of the system are used to generate a length 24 binary code. Any two such codes are then shown to be equivalent up to a permutation of the coordinates. This code turns out to be the extended Golay code.In the uniqueness proof for the (4, 7, 23) system, the blocks generate a length 23 code which is extended to a length 24 code. The minimum weight vectors of this larger code hold a (5, 8, 24) Steiner system. This result together with the previous one completes the proof. At this point it is also possible to conclude that the codes involved are unique and hence equivalent to the binary perfect Golay code and its extension.Continuing with the uniqueness result for the (3, 6, 22) Steiner system, the blocks generate a length 22 code which is extended to the same length 24 code by the addition of two coordinates and one additional vector. This extension ultimately requires the computation of the coset weight distribution of the length 22 code, a result heretofore unknown. The complete coset weight distribution for a specific (22, 11, 6) self-dual code is computed using the CAMAC computer system.The (5, 6, 12) and (4, 5, 11) Steiner systems are treated differently. It is shown that each system is completely determined by the choice of six blocks which may be assumed to lie in any such design. These six blocks in fact form a basis for length 12 (and 11) ternary codes corresponding to the two systems and may be generated by an algorithm independent of the designs. This algorithm is presented and the minimum weight vectors of the resulting codes, the perfect ternary Golay code and its extension, are calculated by the CAMAC system.