Best Known (180, 180+20, s)-Nets in Base 4
(180, 180+20, 838894)-Net over F4 — Constructive and digital
Digital (180, 200, 838894)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 20, 34)-net over F4, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 17)-net over F16, using
- digital (160, 180, 838860)-net over F4, using
- net defined by OOA [i] based on linear OOA(4180, 838860, F4, 20, 20) (dual of [(838860, 20), 16777020, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4180, 8388600, F4, 20) (dual of [8388600, 8388420, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4180, 8388600, F4, 20) (dual of [8388600, 8388420, 21]-code), using
- net defined by OOA [i] based on linear OOA(4180, 838860, F4, 20, 20) (dual of [(838860, 20), 16777020, 21]-NRT-code), using
- digital (10, 20, 34)-net over F4, using
(180, 180+20, large)-Net over F4 — Digital
Digital (180, 200, large)-net over F4, using
- 43 times duplication [i] based on digital (177, 197, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 17 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4197, large, F4, 20) (dual of [large, large−197, 21]-code), using
(180, 180+20, large)-Net in Base 4 — Upper bound on s
There is no (180, 200, large)-net in base 4, because
- 18 times m-reduction [i] would yield (180, 182, large)-net in base 4, but