Best Known (136, 161, s)-Nets in Base 3
(136, 161, 4920)-Net over F3 — Constructive and digital
Digital (136, 161, 4920)-net over F3, using
- net defined by OOA [i] based on linear OOA(3161, 4920, F3, 25, 25) (dual of [(4920, 25), 122839, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3161, 59041, F3, 25) (dual of [59041, 58880, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3161, 59041, F3, 25) (dual of [59041, 58880, 26]-code), using
(136, 161, 16855)-Net over F3 — Digital
Digital (136, 161, 16855)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3161, 16855, F3, 3, 25) (dual of [(16855, 3), 50404, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 19683, F3, 3, 25) (dual of [(19683, 3), 58888, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- OOA 3-folding [i] based on linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(3161, 19683, F3, 3, 25) (dual of [(19683, 3), 58888, 26]-NRT-code), using
(136, 161, 6080612)-Net in Base 3 — Upper bound on s
There is no (136, 161, 6080613)-net in base 3, because
- 1 times m-reduction [i] would yield (136, 160, 6080613)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21847 472641 872107 068520 706956 490943 387397 215672 446968 411762 633614 680746 583089 > 3160 [i]