Best Known (128, 128+37, s)-Nets in Base 4
(128, 128+37, 1044)-Net over F4 — Constructive and digital
Digital (128, 165, 1044)-net over F4, using
- 41 times duplication [i] based on digital (127, 164, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 41, 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, 41, 261)-net over F256, using
(128, 128+37, 3043)-Net over F4 — Digital
Digital (128, 165, 3043)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4165, 3043, F4, 37) (dual of [3043, 2878, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4165, 4105, F4, 37) (dual of [4105, 3940, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(34) ⊂ 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(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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, 8, F4, 1) (dual of [8, 7, 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(36) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4165, 4105, F4, 37) (dual of [4105, 3940, 38]-code), using
(128, 128+37, 769903)-Net in Base 4 — Upper bound on s
There is no (128, 165, 769904)-net in base 4, because
- 1 times m-reduction [i] would yield (128, 164, 769904)-net in base 4, but
- 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]