Best Known (157−35, 157, s)-Nets in Base 4
(157−35, 157, 1044)-Net over F4 — Constructive and digital
Digital (122, 157, 1044)-net over F4, using
- 41 times duplication [i] based on digital (121, 156, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 39, 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, 39, 261)-net over F256, using
(157−35, 157, 3053)-Net over F4 — Digital
Digital (122, 157, 3053)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 3053, F4, 35) (dual of [3053, 2896, 36]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using
(157−35, 157, 800944)-Net in Base 4 — Upper bound on s
There is no (122, 157, 800945)-net in base 4, because
- 1 times m-reduction [i] would yield (122, 156, 800945)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 8343 813702 197619 287686 228092 294441 306415 308844 074418 003003 532930 714279 900359 095481 039087 613692 > 4156 [i]