Best Known (86−20, 86, s)-Nets in Base 5
(86−20, 86, 356)-Net over F5 — Constructive and digital
Digital (66, 86, 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 (40, 60, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 30, 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, 30, 126)-net over F25, using
- digital (16, 26, 104)-net over F5, using
(86−20, 86, 3231)-Net over F5 — Digital
Digital (66, 86, 3231)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 3231, F5, 20) (dual of [3231, 3145, 21]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 54 times 0) [i] based on linear OA(580, 3125, F5, 20) (dual of [3125, 3045, 21]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 54 times 0) [i] based on linear OA(580, 3125, F5, 20) (dual of [3125, 3045, 21]-code), using
(86−20, 86, 1161597)-Net in Base 5 — Upper bound on s
There is no (66, 86, 1161598)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 292473 381826 746010 092739 290819 448010 776353 439276 136196 844161 > 586 [i]