Best Known (130, 161, s)-Nets in Base 3
(130, 161, 688)-Net over F3 — Constructive and digital
Digital (130, 161, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (130, 164, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 41, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 41, 172)-net over F81, using
(130, 161, 2866)-Net over F3 — Digital
Digital (130, 161, 2866)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3161, 2866, F3, 2, 31) (dual of [(2866, 2), 5571, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 3281, F3, 2, 31) (dual of [(3281, 2), 6401, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 3281, F3, 2, 31) (dual of [(3281, 2), 6401, 32]-NRT-code), using
(130, 161, 394470)-Net in Base 3 — Upper bound on s
There is no (130, 161, 394471)-net in base 3, because
- 1 times m-reduction [i] would yield (130, 160, 394471)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21848 216414 980789 257922 267799 833489 013179 634911 703292 721214 011855 845193 020291 > 3160 [i]