Best Known (157, 157+18, s)-Nets in Base 4
(157, 157+18, 932101)-Net over F4 — Constructive and digital
Digital (157, 175, 932101)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 9, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- trace code for nets [i] based on digital (0, 9, 17)-net over F16, using
- digital (139, 157, 932067)-net over F4, using
- net defined by OOA [i] based on linear OOA(4157, 932067, F4, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- net defined by OOA [i] based on linear OOA(4157, 932067, F4, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
- digital (9, 18, 34)-net over F4, using
(157, 157+18, 7995014)-Net over F4 — Digital
Digital (157, 175, 7995014)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4175, 7995014, F4, 18) (dual of [7995014, 7994839, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4175, large, F4, 18) (dual of [large, large−175, 19]-code), using
- 18 times code embedding in larger space [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 18 times code embedding in larger space [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4175, large, F4, 18) (dual of [large, large−175, 19]-code), using
(157, 157+18, large)-Net in Base 4 — Upper bound on s
There is no (157, 175, large)-net in base 4, because
- 16 times m-reduction [i] would yield (157, 159, large)-net in base 4, but