Best Known (163−26, 163, s)-Nets in Base 2
(163−26, 163, 480)-Net over F2 — Constructive and digital
Digital (137, 163, 480)-net over F2, using
- 21 times duplication [i] based on digital (136, 162, 480)-net over F2, using
- trace code for nets [i] based on digital (1, 27, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 27, 80)-net over F64, using
(163−26, 163, 1368)-Net over F2 — Digital
Digital (137, 163, 1368)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2163, 1368, F2, 3, 26) (dual of [(1368, 3), 3941, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2163, 1375, F2, 3, 26) (dual of [(1375, 3), 3962, 27]-NRT-code), using
- strength reduction [i] based on linear OOA(2163, 1375, F2, 3, 27) (dual of [(1375, 3), 3962, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2163, 4125, F2, 27) (dual of [4125, 3962, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2163, 4126, F2, 27) (dual of [4126, 3963, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2163, 4126, F2, 27) (dual of [4126, 3963, 28]-code), using
- OOA 3-folding [i] based on linear OA(2163, 4125, F2, 27) (dual of [4125, 3962, 28]-code), using
- strength reduction [i] based on linear OOA(2163, 1375, F2, 3, 27) (dual of [(1375, 3), 3962, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2163, 1375, F2, 3, 26) (dual of [(1375, 3), 3962, 27]-NRT-code), using
(163−26, 163, 33698)-Net in Base 2 — Upper bound on s
There is no (137, 163, 33699)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11 692903 259950 891941 396190 938182 636500 345981 381888 > 2163 [i]