Best Known (171, 171+29, s)-Nets in Base 3
(171, 171+29, 4220)-Net over F3 — Constructive and digital
Digital (171, 200, 4220)-net over F3, using
- 33 times duplication [i] based on digital (168, 197, 4220)-net over F3, using
- net defined by OOA [i] based on linear OOA(3197, 4220, F3, 29, 29) (dual of [(4220, 29), 122183, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3197, 59081, F3, 29) (dual of [59081, 58884, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, 59085, F3, 29) (dual of [59085, 58888, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3197, 59085, F3, 29) (dual of [59085, 58888, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3197, 59081, F3, 29) (dual of [59081, 58884, 30]-code), using
- net defined by OOA [i] based on linear OOA(3197, 4220, F3, 29, 29) (dual of [(4220, 29), 122183, 30]-NRT-code), using
(171, 171+29, 22659)-Net over F3 — Digital
Digital (171, 200, 22659)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3200, 22659, F3, 2, 29) (dual of [(22659, 2), 45118, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3200, 29544, F3, 2, 29) (dual of [(29544, 2), 58888, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3198, 29543, F3, 2, 29) (dual of [(29543, 2), 58888, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3198, 59086, F3, 29) (dual of [59086, 58888, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3197, 59085, F3, 29) (dual of [59085, 58888, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3191, 59049, F3, 29) (dual of [59049, 58858, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3197, 59085, F3, 29) (dual of [59085, 58888, 30]-code), using
- OOA 2-folding [i] based on linear OA(3198, 59086, F3, 29) (dual of [59086, 58888, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3198, 29543, F3, 2, 29) (dual of [(29543, 2), 58888, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3200, 29544, F3, 2, 29) (dual of [(29544, 2), 58888, 30]-NRT-code), using
(171, 171+29, large)-Net in Base 3 — Upper bound on s
There is no (171, 200, large)-net in base 3, because
- 27 times m-reduction [i] would yield (171, 173, large)-net in base 3, but