Best Known (69, 69+20, s)-Nets in Base 5
(69, 69+20, 384)-Net over F5 — Constructive and digital
Digital (69, 89, 384)-net over F5, using
- 1 times m-reduction [i] based on digital (69, 90, 384)-net over F5, using
- (u, u+v)-construction [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
- 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 (18, 28, 132)-net over F5, using
- (u, u+v)-construction [i] based on
(69, 69+20, 3765)-Net over F5 — Digital
Digital (69, 89, 3765)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(589, 3765, F5, 20) (dual of [3765, 3676, 21]-code), using
- 631 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 54 times 0, 1, 104 times 0, 1, 175 times 0, 1, 249 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
- 631 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0, 1, 54 times 0, 1, 104 times 0, 1, 175 times 0, 1, 249 times 0) [i] based on linear OA(580, 3125, F5, 20) (dual of [3125, 3045, 21]-code), using
(69, 69+20, 1882555)-Net in Base 5 — Upper bound on s
There is no (69, 89, 1882556)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 161 559339 229153 946334 039235 978976 318317 461980 252779 275282 634305 > 589 [i]