Best Known (88−14, 88, s)-Nets in Base 3
(88−14, 88, 2816)-Net over F3 — Constructive and digital
Digital (74, 88, 2816)-net over F3, using
- net defined by OOA [i] based on linear OOA(388, 2816, F3, 14, 14) (dual of [(2816, 14), 39336, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(388, 19712, F3, 14) (dual of [19712, 19624, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-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(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(388, 19712, F3, 14) (dual of [19712, 19624, 15]-code), using
(88−14, 88, 9858)-Net over F3 — Digital
Digital (74, 88, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(388, 9858, F3, 2, 14) (dual of [(9858, 2), 19628, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-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(13) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
(88−14, 88, 1682607)-Net in Base 3 — Upper bound on s
There is no (74, 88, 1682608)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 969776 873564 230381 857121 701868 914770 928705 > 388 [i]