Best Known (29−8, 29, s)-Nets in Base 3
(29−8, 29, 114)-Net over F3 — Constructive and digital
Digital (21, 29, 114)-net over F3, using
- 1 times m-reduction [i] based on digital (21, 30, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 10, 38)-net over F27, using
(29−8, 29, 247)-Net over F3 — Digital
Digital (21, 29, 247)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(329, 247, F3, 8) (dual of [247, 218, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(329, 256, F3, 8) (dual of [256, 227, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(316, 243, F3, 5) (dual of [243, 227, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(329, 256, F3, 8) (dual of [256, 227, 9]-code), using
(29−8, 29, 3181)-Net in Base 3 — Upper bound on s
There is no (21, 29, 3182)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 68 646484 557577 > 329 [i]