Best Known (58, 77, s)-Nets in Base 3
(58, 77, 328)-Net over F3 — Constructive and digital
Digital (58, 77, 328)-net over F3, using
- 31 times duplication [i] based on digital (57, 76, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 19, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 19, 82)-net over F81, using
(58, 77, 473)-Net over F3 — Digital
Digital (58, 77, 473)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(377, 473, F3, 19) (dual of [473, 396, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(377, 740, F3, 19) (dual of [740, 663, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(373, 730, F3, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(361, 730, F3, 15) (dual of [730, 669, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(34, 10, F3, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(377, 740, F3, 19) (dual of [740, 663, 20]-code), using
(58, 77, 22160)-Net in Base 3 — Upper bound on s
There is no (58, 77, 22161)-net in base 3, because
- 1 times m-reduction [i] would yield (58, 76, 22161)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 825197 108805 408021 292276 021770 465027 > 376 [i]