Best Known (108−14, 108, s)-Nets in Base 3
(108−14, 108, 25315)-Net over F3 — Constructive and digital
Digital (94, 108, 25315)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (86, 100, 25308)-net over F3, using
- net defined by OOA [i] based on linear OOA(3100, 25308, F3, 14, 14) (dual of [(25308, 14), 354212, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(3100, 177156, F3, 14) (dual of [177156, 177056, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(3100, 177158, F3, 14) (dual of [177158, 177058, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3100, 177158, F3, 14) (dual of [177158, 177058, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(3100, 177156, F3, 14) (dual of [177156, 177056, 15]-code), using
- net defined by OOA [i] based on linear OOA(3100, 25308, F3, 14, 14) (dual of [(25308, 14), 354212, 15]-NRT-code), using
- digital (1, 8, 7)-net over F3, using
(108−14, 108, 88594)-Net over F3 — Digital
Digital (94, 108, 88594)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3108, 88594, F3, 2, 14) (dual of [(88594, 2), 177080, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 88593, F3, 2, 14) (dual of [(88593, 2), 177080, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 177186, F3, 14) (dual of [177186, 177080, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(3100, 177147, F3, 14) (dual of [177147, 177047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-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(3106, 177186, F3, 14) (dual of [177186, 177080, 15]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 88593, F3, 2, 14) (dual of [(88593, 2), 177080, 15]-NRT-code), using
(108−14, 108, large)-Net in Base 3 — Upper bound on s
There is no (94, 108, large)-net in base 3, because
- 12 times m-reduction [i] would yield (94, 96, large)-net in base 3, but