What is an LD Specialist?

LD design is a kind of auxiliary design, which was introduced by China's Lu Jiayu in solving the Steiner ternary set problem. This type of design has an important effect on the Steiner ternary set problem. Sexual problems are also useful. The existence of LD (n) has been basically solved. Except for a dozen n values, LD (n) exists ^[1] .

Let X be an n-ary set, and L¹ and L² are orthogonal tables OA (n, 4) on X respectively. For each xX, L _x is an orthogonal table OA (n-1 on X \ {x} , 3) (see "Orthogonal Array" below). If c X, make (x, x, x, c) L¹L² for any x X, and for any (u, v, w) X³, or x X such that (u , V, w) L _x , or the existence of t _X and j {1, 2} such that (u, v, w, t) L ^{j is} called {L¹, L²} {L _x | x X} is the LD design and is denoted as LD (n). This type of design has an important effect on the large set problem of Steiner ternary system, and it is also useful for the existence of other combined designs. ) All exist ^[1]

Orthogonal arrays are orthogonal tables. Orthogonal arrays are a type of combination design. Let A be the v × k matrix on the v-element set X. For a sub-matrix composed of any d (2dk) columns, X Each d-element permutation on the row as a submatrix appears times, then A is called an orthogonal array of size N, number of constraints k, number of levels v, intensity d, and exponent . It is called an orthogonal table in the experimental design. , Is denoted as OA (N, k, v, d), and an orthogonal array with intensity 2 defined as N = v ^{d is} denoted as OA (v, k; ). When = 1, it is abbreviated as OA (v, ^d ; , K). The existence of OA (v, k; ) is equivalent to the existence of the cross-section design TD [k; v], and the existence of OA (v, k) is equivalent to k-2 v Existence of first order mutual orthogonal Latin squares ^[1]

Research on Steiner Trilogy

Lu Jiazheng is the only ordinary middle school teacher who won the first prize of the National Natural Science Award. The Steiner ternary problem he solved belongs to the field of design theory of combinatorial mathematics. It was proposed by Cayley, Sylvester, and Kirkman in the 1850s. It has important significance in the theory and application of combinatorial design. The difficulty itself, progress was very small until the 1970s. Beginning in 1972, some foreign design designers introduced the backward push method, which partially promoted it, but until 1980, the results were still fragmentary, far from being completely resolved. After more than 20 years of hard research, Lu Jiazhen finally solved this problem from 1981 to 1983, and gave a solution framework for the remaining 6 undefined values. The International Journal of Combined Mathematics published six articles written by him in September 1981 and April 1983 to solve this problem. This achievement won the first prize of the third National Natural Science Award in 1987. ^[2]

From 1981 to 1983, Lu Jiayu rigorously laid out the overall problem-solving pattern with 16 lemmas and 29 theorems, and introduced a recursive structure. He created a series of auxiliary designs, and used various conclusions that the predecessors had. Complex calculations and inductions finally prove the large set theorem:

If v1,3 (mod 6), v> 7, and v

{141, 283, 501, 789, 1 501, 2 305}, then D (v) = v-2, people also compiled the algorithm program of the large set theorem, and verified on computer that the conclusion of the large set theorem is completely correct of.

This achievement of Lu Jiazheng has been included in the history of combinatorial mathematics. ^[3]