Best Known (22, 22+27, s)-Nets in Base 16
(22, 22+27, 103)-Net over F16 — Constructive and digital
Digital (22, 49, 103)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (6, 33, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (3, 16, 38)-net over F16, using
(22, 22+27, 131)-Net over F16 — Digital
Digital (22, 49, 131)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1649, 131, F16, 2, 27) (dual of [(131, 2), 213, 28]-NRT-code), using
- construction X applied to AG(2;F,228P) ⊂ AG(2;F,232P) [i] based on
- linear OOA(1646, 128, F16, 2, 27) (dual of [(128, 2), 210, 28]-NRT-code), using algebraic-geometric NRT-code AG(2;F,228P) [i] based on function field F/F16 with g(F) = 19 and N(F) ≥ 129, using
- linear OOA(1642, 128, F16, 2, 23) (dual of [(128, 2), 214, 24]-NRT-code), using algebraic-geometric NRT-code AG(2;F,232P) [i] based on function field F/F16 with g(F) = 19 and N(F) ≥ 129 (see above)
- linear OOA(163, 3, F16, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(163, 16, F16, 2, 3) (dual of [(16, 2), 29, 4]-NRT-code), using
- Reed–Solomon NRT-code RS(2;29,16) [i]
- discarding factors / shortening the dual code based on linear OOA(163, 16, F16, 2, 3) (dual of [(16, 2), 29, 4]-NRT-code), using
- construction X applied to AG(2;F,228P) ⊂ AG(2;F,232P) [i] based on
(22, 22+27, 150)-Net in Base 16 — Constructive
(22, 49, 150)-net in base 16, using
- base change [i] based on digital (1, 28, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
(22, 22+27, 10544)-Net in Base 16 — Upper bound on s
There is no (22, 49, 10545)-net in base 16, because
- 1 times m-reduction [i] would yield (22, 48, 10545)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 6283 277606 214941 123124 172021 038492 931742 963786 970898 961776 > 1648 [i]