Best Known (164−36, 164, s)-Nets in Base 4
(164−36, 164, 1048)-Net over F4 — Constructive and digital
Digital (128, 164, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 41, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(164−36, 164, 3446)-Net over F4 — Digital
Digital (128, 164, 3446)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4164, 3446, F4, 36) (dual of [3446, 3282, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, 4109, F4, 36) (dual of [4109, 3945, 37]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4151, 4096, F4, 34) (dual of [4096, 3945, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4164, 4109, F4, 36) (dual of [4109, 3945, 37]-code), using
(164−36, 164, 769903)-Net in Base 4 — Upper bound on s
There is no (128, 164, 769904)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 546 823483 672146 904798 051933 226073 195695 050511 171706 727303 065930 151256 951825 929485 209477 341702 297678 > 4164 [i]