Best Known (37, 37+13, s)-Nets in Base 3
(37, 37+13, 164)-Net over F3 — Constructive and digital
Digital (37, 50, 164)-net over F3, using
- trace code for nets [i] based on digital (12, 25, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(12,81) in PG(24,9)) 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
- base reduction for projective spaces (embedding PG(12,81) in PG(24,9)) for nets [i] based on digital (0, 13, 82)-net over F81, using
(37, 37+13, 368)-Net over F3 — Digital
Digital (37, 50, 368)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(350, 368, F3, 2, 13) (dual of [(368, 2), 686, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(350, 736, F3, 13) (dual of [736, 686, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(349, 729, F3, 13) (dual of [729, 680, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(343, 729, F3, 11) (dual of [729, 686, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(350, 736, F3, 13) (dual of [736, 686, 14]-code), using
(37, 37+13, 11789)-Net in Base 3 — Upper bound on s
There is no (37, 50, 11790)-net in base 3, because
- 1 times m-reduction [i] would yield (37, 49, 11790)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 239411 699320 512359 725957 > 349 [i]