Best Known (54−20, 54, s)-Nets in Base 5
(54−20, 54, 132)-Net over F5 — Constructive and digital
Digital (34, 54, 132)-net over F5, using
- 6 times m-reduction [i] based on digital (34, 60, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 30, 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, 30, 66)-net over F25, using
(54−20, 54, 202)-Net over F5 — Digital
Digital (34, 54, 202)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(554, 202, F5, 20) (dual of [202, 148, 21]-code), using
- 147 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0) [i] based on linear OA(520, 21, F5, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,5)), using
- dual of repetition code with length 21 [i]
- 147 step Varšamov–Edel lengthening with (ri) = (5, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0) [i] based on linear OA(520, 21, F5, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,5)), using
(54−20, 54, 6728)-Net in Base 5 — Upper bound on s
There is no (34, 54, 6729)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 55 571986 981125 766649 869141 525657 807865 > 554 [i]