Best Known (39, 52, s)-Nets in Base 3
(39, 52, 328)-Net over F3 — Constructive and digital
Digital (39, 52, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 13, 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
(39, 52, 391)-Net over F3 — Digital
Digital (39, 52, 391)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(352, 391, F3, 13) (dual of [391, 339, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(352, 734, F3, 13) (dual of [734, 682, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(349, 730, F3, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(337, 730, F3, 9) (dual of [730, 693, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(352, 734, F3, 13) (dual of [734, 682, 14]-code), using
(39, 52, 17005)-Net in Base 3 — Upper bound on s
There is no (39, 52, 17006)-net in base 3, because
- 1 times m-reduction [i] would yield (39, 51, 17006)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 154284 318440 981464 786885 > 351 [i]