Best Known (67, 67+32, s)-Nets in Base 5
(67, 67+32, 252)-Net over F5 — Constructive and digital
Digital (67, 99, 252)-net over F5, using
- 15 times m-reduction [i] based on digital (67, 114, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 57, 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, 57, 126)-net over F25, using
(67, 67+32, 559)-Net over F5 — Digital
Digital (67, 99, 559)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(599, 559, F5, 32) (dual of [559, 460, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(599, 624, F5, 32) (dual of [624, 525, 33]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(599, 624, F5, 32) (dual of [624, 525, 33]-code), using
(67, 67+32, 35910)-Net in Base 5 — Upper bound on s
There is no (67, 99, 35911)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1578 355629 533580 952273 869506 226531 389152 441829 307027 491417 709586 521025 > 599 [i]