Best Known (52−29, 52, s)-Nets in Base 16
(52−29, 52, 103)-Net over F16 — Constructive and digital
Digital (23, 52, 103)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 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, 35, 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, 17, 38)-net over F16, using
(52−29, 52, 129)-Net in Base 16 — Constructive
(23, 52, 129)-net in base 16, using
- 161 times duplication [i] based on (22, 51, 129)-net in base 16, using
- base change [i] based on (5, 34, 129)-net in base 64, using
- 1 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- base change [i] based on digital (0, 30, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 30, 129)-net over F128, using
- 1 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- base change [i] based on (5, 34, 129)-net in base 64, using
(52−29, 52, 132)-Net over F16 — Digital
Digital (23, 52, 132)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1652, 132, F16, 2, 29) (dual of [(132, 2), 212, 30]-NRT-code), using
- construction X applied to AG(2;F,226P) ⊂ AG(2;F,231P) [i] based on
- linear OOA(1648, 128, F16, 2, 29) (dual of [(128, 2), 208, 30]-NRT-code), using algebraic-geometric NRT-code AG(2;F,226P) [i] based on function field F/F16 with g(F) = 19 and N(F) ≥ 129, using
- linear OOA(1643, 128, F16, 2, 24) (dual of [(128, 2), 213, 25]-NRT-code), using algebraic-geometric NRT-code AG(2;F,231P) [i] based on function field F/F16 with g(F) = 19 and N(F) ≥ 129 (see above)
- linear OOA(164, 4, F16, 2, 4) (dual of [(4, 2), 4, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(164, 16, F16, 2, 4) (dual of [(16, 2), 28, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(2;28,16) [i]
- discarding factors / shortening the dual code based on linear OOA(164, 16, F16, 2, 4) (dual of [(16, 2), 28, 5]-NRT-code), using
- construction X applied to AG(2;F,226P) ⊂ AG(2;F,231P) [i] based on
(52−29, 52, 133)-Net in Base 16
(23, 52, 133)-net in base 16, using
- 2 times m-reduction [i] based on (23, 54, 133)-net in base 16, using
- base change [i] based on digital (5, 36, 133)-net over F64, using
- net from sequence [i] based on digital (5, 132)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 133, using
- net from sequence [i] based on digital (5, 132)-sequence over F64, using
- base change [i] based on digital (5, 36, 133)-net over F64, using
(52−29, 52, 9805)-Net in Base 16 — Upper bound on s
There is no (23, 52, 9806)-net in base 16, because
- 1 times m-reduction [i] would yield (23, 51, 9806)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 25 725393 354562 196151 554017 797831 146968 552057 932270 335940 852736 > 1651 [i]