An \(N \times k\) array A with entries from v-set \({\mathcal {V}}\) is said to be an orthogonal array with v levels, strength t and index \(\lambda \), denoted by OA(N; t, k, v), if every \(N \times t\) sub-array of A contains each t-tuple based on \({\mathcal {V}}\) exactly \(\lambda \) times as a row. An OA(N; t, k, v) is called irredundant, denoted by IrOA(N; t, k, v), if in any \(N\times (k-t )\) sub-array, all of its rows are different. The definition of an IrOA was firstly introduced by Goyeneche and \({\dot{Z}}\)yczkowski (Phys Rev A 90:022316, 2014) who showed an IrOA(N; t, k, v) corresponds to a t-uniform state of k subsystems with local dimension v. In this paper, we construct some kinds of 2-uniform states by establishing the existence of an IrOA\((v^3;2,12,v)\) for any integer \((v\ge 4)\) and \((v\not \equiv 2\pmod 4)\), and an IrOA\((v^3;2,3v,v)\) for any prime or prime power \(v\ge 3\).

带有v -set \（{{mathcal {V}} \）中的项的\（N \ times k \）数组A称为具有v级，强度t和索引\（\ lambda \）的正交数组，由OA表示为（ñ ; 吨，  ķ，  v），如果每\（N \时间t \）的子阵列甲包含每个吨基于元组\（{{v} \ mathcal} \）恰好\（\连续\）次。OA（N ;  t，  k， v）称为irredundant，用IrOA（N ;  t，  k，  v）表示，如果在任何\（N \ times（kt）\）子数组中，其所有行都是不同的。IrOA的定义最早由Goyeneche和\（{\ dot {Z}} \） yczkowski（Phys Rev A 90：022316，2014）提出，他们指出IrOA（N ;  t，  k，  v）对应于t具有局部维v的k个子系统的一致状态。在本文中，我们通过建立IrOA的存在来构造某些2均匀状态\（（v ^ 3; 2,12，v）\）对于任何整数\ {{v \ ge 4} \}和\ {{v \ not \ equiv 2 \ pmod 4} \）和IrOA \ { （v ^ 3; 2,3v，v）\）对于任何素数或素数幂（（v \ ge 3 \））。