Best Known (71, 71+19, s)-Nets in Base 4
(71, 71+19, 1045)-Net over F4 — Constructive and digital
Digital (71, 90, 1045)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (57, 76, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 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
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- digital (5, 14, 17)-net over F4, using
(71, 71+19, 3382)-Net over F4 — Digital
Digital (71, 90, 3382)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(490, 3382, F4, 19) (dual of [3382, 3292, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(490, 4119, F4, 19) (dual of [4119, 4029, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(467, 4096, F4, 15) (dual of [4096, 4029, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(490, 4119, F4, 19) (dual of [4119, 4029, 20]-code), using
(71, 71+19, 1242602)-Net in Base 4 — Upper bound on s
There is no (71, 90, 1242603)-net in base 4, because
- 1 times m-reduction [i] would yield (71, 89, 1242603)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 383126 205475 793569 227866 204221 430157 731586 339851 753914 > 489 [i]