Best Known (163, 163+23, s)-Nets in Base 4
(163, 163+23, 95340)-Net over F4 — Constructive and digital
Digital (163, 186, 95340)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 15, 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 (148, 171, 95325)-net over F4, using
- net defined by OOA [i] based on linear OOA(4171, 95325, F4, 23, 23) (dual of [(95325, 23), 2192304, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 11-folding and stacking with additional row [i] based on linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using
- net defined by OOA [i] based on linear OOA(4171, 95325, F4, 23, 23) (dual of [(95325, 23), 2192304, 24]-NRT-code), using
- digital (4, 15, 15)-net over F4, using
(163, 163+23, 582399)-Net over F4 — Digital
Digital (163, 186, 582399)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4186, 582399, F4, 23) (dual of [582399, 582213, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 1048592, F4, 23) (dual of [1048592, 1048406, 24]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- dual of repetition code with length 8 [i]
- linear OA(48, 8, F4, 8) (dual of [8, 0, 9]-code or 8-arc in PG(7,4)), using
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(4186, 1048592, F4, 23) (dual of [1048592, 1048406, 24]-code), using
(163, 163+23, large)-Net in Base 4 — Upper bound on s
There is no (163, 186, large)-net in base 4, because
- 21 times m-reduction [i] would yield (163, 165, large)-net in base 4, but