Best Known (89−29, 89, s)-Nets in Base 5
(89−29, 89, 252)-Net over F5 — Constructive and digital
Digital (60, 89, 252)-net over F5, using
- 11 times m-reduction [i] based on digital (60, 100, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 50, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 50, 126)-net over F25, using
(89−29, 89, 485)-Net over F5 — Digital
Digital (60, 89, 485)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 485, F5, 29) (dual of [485, 396, 30]-code), using
- 395 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0) [i] based on linear OA(529, 30, F5, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,5)), using
- dual of repetition code with length 30 [i]
- 395 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0) [i] based on linear OA(529, 30, F5, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,5)), using
(89−29, 89, 37394)-Net in Base 5 — Upper bound on s
There is no (60, 89, 37395)-net in base 5, because
- 1 times m-reduction [i] would yield (60, 88, 37395)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 32 318157 622780 583423 418277 135699 360633 995034 920687 130421 892553 > 588 [i]