Best Known (52−12, 52, s)-Nets in Base 3
(52−12, 52, 400)-Net over F3 — Constructive and digital
Digital (40, 52, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 13, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(52−12, 52, 605)-Net over F3 — Digital
Digital (40, 52, 605)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(352, 605, F3, 12) (dual of [605, 553, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(352, 743, F3, 12) (dual of [743, 691, 13]-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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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, 743, F3, 12) (dual of [743, 691, 13]-code), using
(52−12, 52, 20423)-Net in Base 3 — Upper bound on s
There is no (40, 52, 20424)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 6 462399 826726 763750 702769 > 352 [i]