Best Known (27, 27+11, s)-Nets in Base 5
(27, 27+11, 172)-Net over F5 — Constructive and digital
Digital (27, 38, 172)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 68)-net over F5, using
- digital (17, 28, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 14, 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, 14, 52)-net over F25, using
(27, 27+11, 660)-Net over F5 — Digital
Digital (27, 38, 660)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(538, 660, F5, 11) (dual of [660, 622, 12]-code), using
- 29 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 7 times 0, 1, 16 times 0) [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]
- 29 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 7 times 0, 1, 16 times 0) [i] based on linear OA(533, 626, F5, 11) (dual of [626, 593, 12]-code), using
(27, 27+11, 96858)-Net in Base 5 — Upper bound on s
There is no (27, 38, 96859)-net in base 5, because
- 1 times m-reduction [i] would yield (27, 37, 96859)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 72 759796 313434 262275 494749 > 537 [i]