Masaaki Harada
In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II $\mathbb{Z}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II $\mathbb{Z}_4$-codes of length $40$.
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By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs with the same parameters as the above 5-designs. We give some parameters of a self-orthogonal 5-design whose existence is equivalent to that of an extremal doubly even self-dual code of length 24m for m=3,...,6. If $m \in \{1,\ldots,6\}$, $k \in \{m+1,\ldots,5m-1\}$ and $(m,k) \ne (6,18)$, then it is shown that an extremal doubly even self-dual code of leng…
In this note, we complete the classification of extremal doubly even self-dual codes with 2-transitive automorphism groups.
Let D be the support design of the minimum weight of an extremal binary doubly even self-dual [24m,12m,4m+4] code. In this note, we consider the case when D becomes a t-design with t \geq 6.
Let C be an extremal binary doubly even self-dual code of length n and s(C) denote the largest integer t such that the support design of C holds a t-design for some weight. In this paper, we prove s(C) \leq 7.
Support of extremal doubly stochastic arrays
本文研究极值$n \times m$双随机阵列的支撑集问题，刻画了其支撑集的可能规模，证明支撑集最小规模为$n+m-\gcd(n,m)$，并给出$m=kn+1$时极值阵列支撑集的结构特征。
A new lower bound for doubly metric dimension and related extremal differences
本文提出基于最小击中集的新图不变量，证明其为双重度量维数的紧下界，并给出若干图类的精确值及极值差异。
In this note, we construct new doubly even self-dual codes having minimum weight $20$ for lengths $112$, $120$ and $128$. This implies that there are at least three inequivalent extremal doubly even self-dual codes of length $112$.
A complete classification of binary doubly even self-dual codes of length 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given.
A relationship between $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes and extremal doubly even self-dual $[24k+8,12k+4,4k+4]$ codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, $s$-extremal singly even self-dual $[56,28,10]$ codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length $24k+8$ with covering radius meeting the Delsarte bound for $k \ge 137$. Similarly, we show that there is no extremal doubly even self-dual code of length $24k+16$ with covering r…
For lengths $64$ and $66$, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound.
Doubly stochastic arrays with small support
本文研究双随机矩阵的最小支撑问题，刻画了一般情形下最小支撑与极矩阵的关系，并证明 n、m 互质时二者等价。
In this paper, we determine the dimensions of the residue codes of extremal Type II Z4-codes for lengths 32 and 40. We demonstrate that every binary doubly even self-dual code of length 32 can be realized as the residue code of some extremal Type II Z4-code. It is also shown that there is a unique extremal Type II Z4-code of length 32 whose residue code has the smallest dimension 6 up to equivalence. As a consequence, many new extremal Type II Z4-codes of lengths 32 and 40 are constructed.
In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.
An $s$-extremal optimal unimodular lattice in dimension $52$ is constructed for the first time. This lattice is constructed from a certain self-dual $\mathbb{F}_5$-code by Construction A. In addition, as neighbors of the lattice, two more $s$-extremal optimal unimodular lattices are constructed.
We prove that a putative $[72,36,16]$ code is not the image of linear code over $\ZZ_4$, $\FF_2 + u \FF_2$ or $\FF_2+v\FF_2$, thus proving that the extremal doubly even $[72,36,16]$-binary code cannot have an automorphism group containing a fixed point-free involution. Combining this with the previously proved result by Bouyuklieva that such a code cannot have an automorphism group containing an involution with fixed points, we conclude that the automorphism group of the $[72,36,16]$-code cannot be of even order, leaving 3 and 5 as the only possibilities.
Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31
本文分类了 p=29、31 时具有 p 阶自同构的 2p+2 阶 Hadamard 矩阵，并据此构造计算，发现了若干新的三元及二元近极值自对偶码。
In this work, connected cubic planar bipartite graphs and related binary self-dual codes are studied. Binary self-dual codes of length 16 are obtained by face-vertex incidence matrices of these graphs. By considering their lifts to the ring R_2 new extremal binary self-dual codes of lengths 64 are constructed as Gray images. More precisely, we construct 15 new codes of length 64. Moreover, 10 new codes of length 66 were obtained by applying a building-up construction to the binary codes. Codes with these weight enumerators are constructed for the first time in the literature. The results are t…
We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the classification of extremal doubly even self-dual codes of length 40, we show that a quasi-symmetric 2-(37,9,8) design with intersection numbers 1 and 3 does not exist.
In this paper, new extremal odd unimodular lattices in dimension $36$ are constructed. Some new odd unimodular lattices in dimension $36$ with long shadows are also constructed.
New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.
It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp.\ $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp.\ $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every positive integer $k$. A number of new self-dual $\mathbb{Z}_4$-codes of length $24$ and minimum Lee weight $10$ are constructed using the above characterization. These codes are Type I $\mathbb{Z}_4$-codes having the largest minimum Lee weight and the largest Euclidean weight among all Type I $\mathbb{Z}_4$-codes of that length. In addition, new extremal Type …
In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets F_2, F_2 + uF_2, F_4+uF_4, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear complementary dual codes (LCD codes for short). Using the method, we construct 661 new inequivalent extremal doubly even $[56, 28, 12]$ codes. Furthermore, constructing LCD codes by the method, we improve some of the previously known lower bounds on the largest minimum weights of binary LCD codes of length $n=26,28 \le n \le 40$.
Construction of extremal Type II $\mathbb{Z}_{8}$-codes via doubling method
本文提出倍增方法，从已知极值类型II $\mathbb{Z}_8$-码构造新码，并据此构造出至少十个此前未知的长度为32、类型为$(15,1,1)$的极值类型II $\mathbb{Z}_8$-码，其二元剩余码为最优$[32,15]$二元码。
New extremal Type II $\mathbb{Z}_4$-codes of length 64 by the doubling method
本文基于倍增法开发了一种构造新极值Type II $\mathbb{Z}_4$-码的方法，并成功构造出三个长度为64、类型为$4^{31}2^2$的新极值Type II $\mathbb{Z}_4$-码，其剩余码为新的最佳已知二元码，且部分码字支撑构成自正交1-设计。
We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of fixed degree of an extremal self-dual vertex operator algebra form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus-Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebra…
Maximal proper extremal subset in the crystal basis of a finite-dimensional irreducible module
本文证明了有限型量子群有限维不可约模的晶体基存在唯一极大真极值子集，并猜想其补集由下降Kashiwara算子作用于特定元素所得，且在A型中验证了该猜想。
Extremal graphs for a conjecture on the square energy of graphs
该研究确定了Elphick等人关于图的正、负邻接特征值平方和最小值下界猜想的所有等号情形，通过结合相关引理与矩阵分析完成证明，明确了等号对应的图类。
Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets
本文研究了与普遍最优配置、Kerdock码和极值欧几里得直线集相关的关联方案，并探讨了其与Barnes-Wall晶格的联系。
It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.
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