Best Known (62, 62+29, s)-Nets in Base 5
(62, 62+29, 252)-Net over F5 — Constructive and digital
Digital (62, 91, 252)-net over F5, using
- 13 times m-reduction [i] based on digital (62, 104, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 52, 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, 52, 126)-net over F25, using
(62, 62+29, 566)-Net over F5 — Digital
Digital (62, 91, 566)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(591, 566, F5, 29) (dual of [566, 475, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(591, 624, F5, 29) (dual of [624, 533, 30]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(591, 624, F5, 29) (dual of [624, 533, 30]-code), using
(62, 62+29, 47063)-Net in Base 5 — Upper bound on s
There is no (62, 91, 47064)-net in base 5, because
- 1 times m-reduction [i] would yield (62, 90, 47064)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 807 917226 442637 750061 481028 177381 667707 781693 256815 634695 352705 > 590 [i]