Best Known (158, 158+16, s)-Nets in Base 4
(158, 158+16, 1049089)-Net over F4 — Constructive and digital
Digital (158, 174, 1049089)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (22, 30, 514)-net over F4, using
- trace code for nets [i] based on digital (7, 15, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(7,256) in PG(14,16)) for nets [i] based on digital (0, 8, 257)-net over F256, using
- trace code for nets [i] based on digital (7, 15, 257)-net over F16, using
- digital (128, 144, 1048575)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 1048575, F4, 16, 16) (dual of [(1048575, 16), 16777056, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(4144, 8388600, F4, 16) (dual of [8388600, 8388456, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(4144, 8388600, F4, 16) (dual of [8388600, 8388456, 17]-code), using
- net defined by OOA [i] based on linear OOA(4144, 1048575, F4, 16, 16) (dual of [(1048575, 16), 16777056, 17]-NRT-code), using
- digital (22, 30, 514)-net over F4, using
(158, 158+16, large)-Net over F4 — Digital
Digital (158, 174, large)-net over F4, using
- 2 times m-reduction [i] based on digital (158, 176, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4176, large, F4, 18) (dual of [large, large−176, 19]-code), using
- 19 times code embedding in larger space [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]
- 19 times code embedding in larger space [i] based on linear OA(4157, large, F4, 18) (dual of [large, large−157, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4176, large, F4, 18) (dual of [large, large−176, 19]-code), using
(158, 158+16, large)-Net in Base 4 — Upper bound on s
There is no (158, 174, large)-net in base 4, because
- 14 times m-reduction [i] would yield (158, 160, large)-net in base 4, but