Best Known (136−21, 136, s)-Nets in Base 3
(136−21, 136, 1972)-Net over F3 — Constructive and digital
Digital (115, 136, 1972)-net over F3, using
- 31 times duplication [i] based on digital (114, 135, 1972)-net over F3, using
- net defined by OOA [i] based on linear OOA(3135, 1972, F3, 21, 21) (dual of [(1972, 21), 41277, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3135, 19721, F3, 21) (dual of [19721, 19586, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 19724, F3, 21) (dual of [19724, 19589, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 19724, F3, 21) (dual of [19724, 19589, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3135, 19721, F3, 21) (dual of [19721, 19586, 22]-code), using
- net defined by OOA [i] based on linear OOA(3135, 1972, F3, 21, 21) (dual of [(1972, 21), 41277, 22]-NRT-code), using
(136−21, 136, 9864)-Net over F3 — Digital
Digital (115, 136, 9864)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3136, 9864, F3, 2, 21) (dual of [(9864, 2), 19592, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3136, 19728, F3, 21) (dual of [19728, 19592, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(39, 45, F3, 4) (dual of [45, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3136, 19728, F3, 21) (dual of [19728, 19592, 22]-code), using
(136−21, 136, 6252915)-Net in Base 3 — Upper bound on s
There is no (115, 136, 6252916)-net in base 3, because
- 1 times m-reduction [i] would yield (115, 135, 6252916)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25785 138423 523673 811579 614542 386393 519939 242031 853968 497624 904537 > 3135 [i]