Best Known (42−23, 42, s)-Nets in Base 8
(42−23, 42, 65)-Net over F8 — Constructive and digital
Digital (19, 42, 65)-net over F8, using
- t-expansion [i] based on digital (14, 42, 65)-net over F8, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- net from sequence [i] based on digital (14, 64)-sequence over F8, using
(42−23, 42, 67)-Net over F8 — Digital
Digital (19, 42, 67)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(842, 67, F8, 3, 23) (dual of [(67, 3), 159, 24]-NRT-code), using
- strength reduction [i] based on linear OOA(842, 67, F8, 3, 24) (dual of [(67, 3), 159, 25]-NRT-code), using
- construction X applied to AG(3;F,167P) ⊂ AG(3;F,172P) [i] based on
- linear OOA(838, 64, F8, 3, 24) (dual of [(64, 3), 154, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,167P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- the Suzuki function field over F8 [i]
- linear OOA(833, 64, F8, 3, 19) (dual of [(64, 3), 159, 20]-NRT-code), using algebraic-geometric NRT-code AG(3;F,172P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65 (see above)
- linear OOA(84, 3, F8, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(84, 8, F8, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(3;20,8) [i]
- discarding factors / shortening the dual code based on linear OOA(84, 8, F8, 3, 4) (dual of [(8, 3), 20, 5]-NRT-code), using
- linear OOA(838, 64, F8, 3, 24) (dual of [(64, 3), 154, 25]-NRT-code), using algebraic-geometric NRT-code AG(3;F,167P) [i] based on function field F/F8 with g(F) = 14 and N(F) ≥ 65, using
- construction X applied to AG(3;F,167P) ⊂ AG(3;F,172P) [i] based on
- strength reduction [i] based on linear OOA(842, 67, F8, 3, 24) (dual of [(67, 3), 159, 25]-NRT-code), using
(42−23, 42, 1622)-Net in Base 8 — Upper bound on s
There is no (19, 42, 1623)-net in base 8, because
- 1 times m-reduction [i] would yield (19, 41, 1623)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 10 658719 362927 550699 960021 442790 119192 > 841 [i]