Best Known (69−20, 69, s)-Nets in Base 3
(69−20, 69, 156)-Net over F3 — Constructive and digital
Digital (49, 69, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 23, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
(69−20, 69, 225)-Net over F3 — Digital
Digital (49, 69, 225)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(369, 225, F3, 20) (dual of [225, 156, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(369, 256, F3, 20) (dual of [256, 187, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(366, 243, F3, 20) (dual of [243, 177, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(356, 243, F3, 17) (dual of [243, 187, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(369, 256, F3, 20) (dual of [256, 187, 21]-code), using
(69−20, 69, 4427)-Net in Base 3 — Upper bound on s
There is no (49, 69, 4428)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 835 434180 843707 528526 841146 819049 > 369 [i]