Best Known (27, 27+10, s)-Nets in Base 5
(27, 27+10, 178)-Net over F5 — Constructive and digital
Digital (27, 37, 178)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 46)-net over F5, using
- 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
(27, 27+10, 800)-Net over F5 — Digital
Digital (27, 37, 800)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(537, 800, F5, 10) (dual of [800, 763, 11]-code), using
- 170 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 51 times 0, 1, 95 times 0) [i] based on linear OA(532, 625, F5, 10) (dual of [625, 593, 11]-code), using
- 1 times truncation [i] based on linear OA(533, 626, F5, 11) (dual of [626, 593, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 626 | 58−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(533, 626, F5, 11) (dual of [626, 593, 12]-code), using
- 170 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 17 times 0, 1, 51 times 0, 1, 95 times 0) [i] based on linear OA(532, 625, F5, 10) (dual of [625, 593, 11]-code), using
(27, 27+10, 96858)-Net in Base 5 — Upper bound on s
There is no (27, 37, 96859)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 72 759796 313434 262275 494749 > 537 [i]