Best Known (50, 50+15, s)-Nets in Base 3
(50, 50+15, 400)-Net over F3 — Constructive and digital
Digital (50, 65, 400)-net over F3, using
- 31 times duplication [i] based on digital (49, 64, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 16, 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
- trace code for nets [i] based on digital (1, 16, 100)-net over F81, using
(50, 50+15, 621)-Net over F3 — Digital
Digital (50, 65, 621)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(365, 621, F3, 15) (dual of [621, 556, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(365, 747, F3, 15) (dual of [747, 682, 16]-code), using
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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, 15, F3, 1) (dual of [15, 14, 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
- linear OA(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(15) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(365, 747, F3, 15) (dual of [747, 682, 16]-code), using
(50, 50+15, 38910)-Net in Base 3 — Upper bound on s
There is no (50, 65, 38911)-net in base 3, because
- 1 times m-reduction [i] would yield (50, 64, 38911)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 433949 753727 359624 179146 612723 > 364 [i]