Best Known (68, 89, s)-Nets in Base 5
(68, 89, 356)-Net over F5 — Constructive and digital
Digital (68, 89, 356)-net over F5, using
- 51 times duplication [i] based on digital (67, 88, 356)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (16, 26, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 13, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 13, 52)-net over F25, using
- digital (41, 62, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 31, 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, 31, 126)-net over F25, using
- digital (16, 26, 104)-net over F5, using
- (u, u+v)-construction [i] based on
(68, 89, 3224)-Net over F5 — Digital
Digital (68, 89, 3224)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 3224, F5, 21) (dual of [3224, 3135, 22]-code), using
- 90 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 40 times 0) [i] based on linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 90 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 0, 0, 1, 6 times 0, 1, 12 times 0, 1, 23 times 0, 1, 40 times 0) [i] based on linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using
(68, 89, 1602693)-Net in Base 5 — Upper bound on s
There is no (68, 89, 1602694)-net in base 5, because
- 1 times m-reduction [i] would yield (68, 88, 1602694)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 32 311860 071065 585235 407804 196587 844219 863074 877459 739397 906433 > 588 [i]