Best Known (117−29, 117, s)-Nets in Base 3
(117−29, 117, 328)-Net over F3 — Constructive and digital
Digital (88, 117, 328)-net over F3, using
- 31 times duplication [i] based on digital (87, 116, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 82)-net over F81, using
(117−29, 117, 589)-Net over F3 — Digital
Digital (88, 117, 589)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3117, 589, F3, 29) (dual of [589, 472, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3117, 749, F3, 29) (dual of [749, 632, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3117, 749, F3, 29) (dual of [749, 632, 30]-code), using
(117−29, 117, 27133)-Net in Base 3 — Upper bound on s
There is no (88, 117, 27134)-net in base 3, because
- 1 times m-reduction [i] would yield (88, 116, 27134)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 22 193650 495190 423844 045596 045135 230981 595977 336171 552053 > 3116 [i]