Best Known (144, 144+20, s)-Nets in Base 4
(144, 144+20, 104872)-Net over F4 — Constructive and digital
Digital (144, 164, 104872)-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 (130, 150, 104857)-net over F4, using
- net defined by OOA [i] based on linear OOA(4150, 104857, F4, 20, 20) (dual of [(104857, 20), 2096990, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4150, 1048570, F4, 20) (dual of [1048570, 1048420, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4150, 1048575, F4, 20) (dual of [1048575, 1048425, 21]-code), using
- 1 times truncation [i] based on linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times truncation [i] based on linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4150, 1048575, F4, 20) (dual of [1048575, 1048425, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4150, 1048570, F4, 20) (dual of [1048570, 1048420, 21]-code), using
- net defined by OOA [i] based on linear OOA(4150, 104857, F4, 20, 20) (dual of [(104857, 20), 2096990, 21]-NRT-code), using
- digital (4, 14, 15)-net over F4, using
(144, 144+20, 712833)-Net over F4 — Digital
Digital (144, 164, 712833)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4164, 712833, F4, 20) (dual of [712833, 712669, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, 1048640, F4, 20) (dual of [1048640, 1048476, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(12) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(413, 64, F4, 6) (dual of [64, 51, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- construction X applied to Ce(20) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4164, 1048640, F4, 20) (dual of [1048640, 1048476, 21]-code), using
(144, 144+20, large)-Net in Base 4 — Upper bound on s
There is no (144, 164, large)-net in base 4, because
- 18 times m-reduction [i] would yield (144, 146, large)-net in base 4, but