Best Known (99−31, 99, s)-Nets in Base 5
(99−31, 99, 264)-Net over F5 — Constructive and digital
Digital (68, 99, 264)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- digital (51, 82, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 41, 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, 41, 126)-net over F25, using
- digital (2, 17, 12)-net over F5, using
(99−31, 99, 648)-Net over F5 — Digital
Digital (68, 99, 648)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 648, F5, 31) (dual of [648, 549, 32]-code), using
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0) [i] based on linear OA(595, 625, F5, 31) (dual of [625, 530, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 19 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 10 times 0) [i] based on linear OA(595, 625, F5, 31) (dual of [625, 530, 32]-code), using
(99−31, 99, 59187)-Net in Base 5 — Upper bound on s
There is no (68, 99, 59188)-net in base 5, because
- 1 times m-reduction [i] would yield (68, 98, 59188)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 315 564564 138271 280198 282184 839527 140790 824413 269218 779682 740611 028305 > 598 [i]