Best Known (101, 129, s)-Nets in Base 4
(101, 129, 1044)-Net over F4 — Constructive and digital
Digital (101, 129, 1044)-net over F4, using
- 41 times duplication [i] based on digital (100, 128, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 32, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
- trace code for nets [i] based on digital (4, 32, 261)-net over F256, using
(101, 129, 3217)-Net over F4 — Digital
Digital (101, 129, 3217)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4129, 3217, F4, 28) (dual of [3217, 3088, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4129, 4111, F4, 28) (dual of [4111, 3982, 29]-code), using
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4115, 4096, F4, 26) (dual of [4096, 3981, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4129, 4111, F4, 28) (dual of [4111, 3982, 29]-code), using
(101, 129, 711021)-Net in Base 4 — Upper bound on s
There is no (101, 129, 711022)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 463177 273655 822877 717127 231220 610147 263722 210791 545487 468065 855877 772104 683464 > 4129 [i]