Best Known (169, 169+18, s)-Nets in Base 4
(169, 169+18, 932307)-Net over F4 — Constructive and digital
Digital (169, 187, 932307)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (21, 30, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 10, 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, 10, 80)-net over F64, 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 (21, 30, 240)-net over F4, using
(169, 169+18, large)-Net over F4 — Digital
Digital (169, 187, large)-net over F4, using
- 41 times duplication [i] based on digital (168, 186, large)-net over F4, using
- t-expansion [i] based on digital (167, 186, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4186, large, F4, 19) (dual of [large, large−186, 20]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 17 times code embedding in larger space [i] based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4186, large, F4, 19) (dual of [large, large−186, 20]-code), using
- t-expansion [i] based on digital (167, 186, large)-net over F4, using
(169, 169+18, large)-Net in Base 4 — Upper bound on s
There is no (169, 187, large)-net in base 4, because
- 16 times m-reduction [i] would yield (169, 171, large)-net in base 4, but