Best Known (106−26, 106, s)-Nets in Base 3
(106−26, 106, 328)-Net over F3 — Constructive and digital
Digital (80, 106, 328)-net over F3, using
- 32 times duplication [i] based on digital (78, 104, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 26, 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
- trace code for nets [i] based on digital (0, 26, 82)-net over F81, using
(106−26, 106, 578)-Net over F3 — Digital
Digital (80, 106, 578)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3106, 578, F3, 26) (dual of [578, 472, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3106, 742, F3, 26) (dual of [742, 636, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(3103, 729, F3, 26) (dual of [729, 626, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3106, 742, F3, 26) (dual of [742, 636, 27]-code), using
(106−26, 106, 22004)-Net in Base 3 — Upper bound on s
There is no (80, 106, 22005)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 375 917834 143919 827668 395935 554135 779158 783224 325339 > 3106 [i]