Best Known (141, 141+20, s)-Nets in Base 4
(141, 141+20, 104866)-Net over F4 — Constructive and digital
Digital (141, 161, 104866)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-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 (1, 11, 9)-net over F4, using
(141, 141+20, 565773)-Net over F4 — Digital
Digital (141, 161, 565773)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 565773, F4, 20) (dual of [565773, 565612, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4161, 1048636, F4, 20) (dual of [1048636, 1048475, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [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(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4161, 1048636, F4, 20) (dual of [1048636, 1048475, 21]-code), using
(141, 141+20, large)-Net in Base 4 — Upper bound on s
There is no (141, 161, large)-net in base 4, because
- 18 times m-reduction [i] would yield (141, 143, large)-net in base 4, but