Best Known (91−19, 91, s)-Nets in Base 4
(91−19, 91, 1046)-Net over F4 — Constructive and digital
Digital (72, 91, 1046)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 18)-net over F4, using
- 3 times m-reduction [i] based on digital (6, 18, 18)-net 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 (6, 15, 18)-net over F4, using
(91−19, 91, 3670)-Net over F4 — Digital
Digital (72, 91, 3670)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(491, 3670, F4, 19) (dual of [3670, 3579, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 4121, F4, 19) (dual of [4121, 4030, 20]-code), using
- construction XX applied to Ce(18) ⊂ Ce(14) ⊂ Ce(13) [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(461, 4096, F4, 14) (dual of [4096, 4035, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(18) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(491, 4121, F4, 19) (dual of [4121, 4030, 20]-code), using
(91−19, 91, 1449532)-Net in Base 4 — Upper bound on s
There is no (72, 91, 1449533)-net in base 4, because
- 1 times m-reduction [i] would yield (72, 90, 1449533)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 532498 638395 515280 926515 632639 963538 053557 317483 622752 > 490 [i]