Best Known (18, 18+9, s)-Nets in Base 5
(18, 18+9, 132)-Net over F5 — Constructive and digital
Digital (18, 27, 132)-net over F5, using
- 1 times m-reduction [i] based on digital (18, 28, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 14, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 14, 66)-net over F25, using
(18, 18+9, 221)-Net over F5 — Digital
Digital (18, 27, 221)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(527, 221, F5, 9) (dual of [221, 194, 10]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 14 times 0, 1, 24 times 0, 1, 34 times 0) [i] based on linear OA(520, 125, F5, 9) (dual of [125, 105, 10]-code), using
- a “GraX†code from Grassl’s database [i]
- 89 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 14 times 0, 1, 24 times 0, 1, 34 times 0) [i] based on linear OA(520, 125, F5, 9) (dual of [125, 105, 10]-code), using
(18, 18+9, 19330)-Net in Base 5 — Upper bound on s
There is no (18, 27, 19331)-net in base 5, because
- 1 times m-reduction [i] would yield (18, 26, 19331)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 1 490289 504696 961905 > 526 [i]