Best Known (50, 50+20, s)-Nets in Base 5
(50, 50+20, 258)-Net over F5 — Constructive and digital
Digital (50, 70, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, 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 (0, 10, 6)-net over F5, using
(50, 50+20, 759)-Net over F5 — Digital
Digital (50, 70, 759)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(570, 759, F5, 20) (dual of [759, 689, 21]-code), using
- 128 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 22 times 0, 1, 38 times 0, 1, 51 times 0) [i] based on linear OA(564, 625, F5, 20) (dual of [625, 561, 21]-code), using
- 1 times truncation [i] based on linear OA(565, 626, F5, 21) (dual of [626, 561, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 626 | 58−1, 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(565, 626, F5, 21) (dual of [626, 561, 22]-code), using
- 128 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 10 times 0, 1, 22 times 0, 1, 38 times 0, 1, 51 times 0) [i] based on linear OA(564, 625, F5, 20) (dual of [625, 561, 21]-code), using
(50, 50+20, 88444)-Net in Base 5 — Upper bound on s
There is no (50, 70, 88445)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 8 470585 739999 896970 411917 250422 013677 387082 652153 > 570 [i]