Best Known (68, 68+12, s)-Nets in Base 3
(68, 68+12, 9841)-Net over F3 — Constructive and digital
Digital (68, 80, 9841)-net over F3, using
- net defined by OOA [i] based on linear OOA(380, 9841, F3, 12, 12) (dual of [(9841, 12), 118012, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(380, 59046, F3, 12) (dual of [59046, 58966, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(380, 59046, F3, 12) (dual of [59046, 58966, 13]-code), using
(68, 68+12, 28291)-Net over F3 — Digital
Digital (68, 80, 28291)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 28291, F3, 2, 12) (dual of [(28291, 2), 56502, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(380, 29524, F3, 2, 12) (dual of [(29524, 2), 58968, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(380, 59048, F3, 12) (dual of [59048, 58968, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 59049, F3, 12) (dual of [59049, 58969, 13]-code), using
- OOA 2-folding [i] based on linear OA(380, 59048, F3, 12) (dual of [59048, 58968, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(380, 29524, F3, 2, 12) (dual of [(29524, 2), 58968, 13]-NRT-code), using
(68, 68+12, 3441978)-Net in Base 3 — Upper bound on s
There is no (68, 80, 3441979)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 147 809020 557944 057828 536579 265307 689037 > 380 [i]