Best Known (174, 174+20, s)-Nets in Base 4
(174, 174+20, 838875)-Net over F4 — Constructive and digital
Digital (174, 194, 838875)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, 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 (4, 14, 15)-net over F4, using
(174, 174+20, 7185036)-Net over F4 — Digital
Digital (174, 194, 7185036)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4194, 7185036, F4, 20) (dual of [7185036, 7184842, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, large, F4, 20) (dual of [large, large−194, 21]-code), using
- 14 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]
- 14 times code embedding in larger space [i] based on linear OA(4180, large, F4, 20) (dual of [large, large−180, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, large, F4, 20) (dual of [large, large−194, 21]-code), using
(174, 174+20, large)-Net in Base 4 — Upper bound on s
There is no (174, 194, large)-net in base 4, because
- 18 times m-reduction [i] would yield (174, 176, large)-net in base 4, but