Best Known (91−23, 91, s)-Nets in Base 3
(91−23, 91, 246)-Net over F3 — Constructive and digital
Digital (68, 91, 246)-net over F3, using
- 31 times duplication [i] based on digital (67, 90, 246)-net over F3, using
- trace code for nets [i] based on digital (7, 30, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- trace code for nets [i] based on digital (7, 30, 82)-net over F27, using
(91−23, 91, 463)-Net over F3 — Digital
Digital (68, 91, 463)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(391, 463, F3, 23) (dual of [463, 372, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using
(91−23, 91, 19654)-Net in Base 3 — Upper bound on s
There is no (68, 91, 19655)-net in base 3, because
- 1 times m-reduction [i] would yield (68, 90, 19655)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 728278 301430 362346 753795 844610 886838 475659 > 390 [i]