Best Known (156−34, 156, s)-Nets in Base 4
(156−34, 156, 1048)-Net over F4 — Constructive and digital
Digital (122, 156, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 39, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(156−34, 156, 3490)-Net over F4 — Digital
Digital (122, 156, 3490)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4156, 3490, F4, 34) (dual of [3490, 3334, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4156, 4119, F4, 34) (dual of [4119, 3963, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- 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(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4156, 4119, F4, 34) (dual of [4119, 3963, 35]-code), using
(156−34, 156, 800944)-Net in Base 4 — Upper bound on s
There is no (122, 156, 800945)-net in base 4, because
- 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]