Best Known (86−22, 86, s)-Nets in Base 4
(86−22, 86, 514)-Net over F4 — Constructive and digital
Digital (64, 86, 514)-net over F4, using
- trace code for nets [i] based on digital (21, 43, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(21,256) in PG(42,16)) for nets [i] based on digital (0, 22, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(21,256) in PG(42,16)) for nets [i] based on digital (0, 22, 257)-net over F256, using
(86−22, 86, 987)-Net over F4 — Digital
Digital (64, 86, 987)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(486, 987, F4, 22) (dual of [987, 901, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 1044, F4, 22) (dual of [1044, 958, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(481, 1024, F4, 22) (dual of [1024, 943, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(466, 1024, F4, 18) (dual of [1024, 958, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(45, 20, F4, 3) (dual of [20, 15, 4]-code or 20-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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 1044, F4, 22) (dual of [1044, 958, 23]-code), using
(86−22, 86, 83341)-Net in Base 4 — Upper bound on s
There is no (64, 86, 83342)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 5986 567995 088761 460857 754774 652503 647718 117387 684210 > 486 [i]