Best Known (164, 164+23, s)-Nets in Base 4
(164, 164+23, 95342)-Net over F4 — Constructive and digital
Digital (164, 187, 95342)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 16, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-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 (5, 16, 17)-net over F4, using
(164, 164+23, 622144)-Net over F4 — Digital
Digital (164, 187, 622144)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4187, 622144, F4, 23) (dual of [622144, 621957, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4187, 1048623, F4, 23) (dual of [1048623, 1048436, 24]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4181, 1048577, F4, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4141, 1048577, F4, 19) (dual of [1048577, 1048436, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(46, 46, F4, 3) (dual of [46, 40, 4]-code or 46-cap in PG(5,4)), using
- discarding factors / shortening the dual code based on linear OA(46, 102, F4, 3) (dual of [102, 96, 4]-code or 102-cap in PG(5,4)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4187, 1048623, F4, 23) (dual of [1048623, 1048436, 24]-code), using
(164, 164+23, large)-Net in Base 4 — Upper bound on s
There is no (164, 187, large)-net in base 4, because
- 21 times m-reduction [i] would yield (164, 166, large)-net in base 4, but