Best Known (57, 76, s)-Nets in Base 3
(57, 76, 328)-Net over F3 — Constructive and digital
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
(57, 76, 442)-Net over F3 — Digital
Digital (57, 76, 442)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(376, 442, F3, 19) (dual of [442, 366, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(376, 734, F3, 19) (dual of [734, 658, 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(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(376, 734, F3, 19) (dual of [734, 658, 20]-code), using
(57, 76, 19612)-Net in Base 3 — Upper bound on s
There is no (57, 76, 19613)-net in base 3, because
- 1 times m-reduction [i] would yield (57, 75, 19613)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 608270 091269 379731 216067 049766 821403 > 375 [i]