Best Known (105, 105+20, s)-Nets in Base 3
(105, 105+20, 1971)-Net over F3 — Constructive and digital
Digital (105, 125, 1971)-net over F3, using
- 31 times duplication [i] based on digital (104, 124, 1971)-net over F3, using
- net defined by OOA [i] based on linear OOA(3124, 1971, F3, 20, 20) (dual of [(1971, 20), 39296, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3124, 19710, F3, 20) (dual of [19710, 19586, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3124, 19716, F3, 20) (dual of [19716, 19592, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(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(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3124, 19716, F3, 20) (dual of [19716, 19592, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3124, 19710, F3, 20) (dual of [19710, 19586, 21]-code), using
- net defined by OOA [i] based on linear OOA(3124, 1971, F3, 20, 20) (dual of [(1971, 20), 39296, 21]-NRT-code), using
(105, 105+20, 9858)-Net over F3 — Digital
Digital (105, 125, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3125, 9858, F3, 2, 20) (dual of [(9858, 2), 19591, 21]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3124, 9858, F3, 2, 20) (dual of [(9858, 2), 19592, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3124, 19716, F3, 20) (dual of [19716, 19592, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- linear OA(3118, 19683, F3, 20) (dual of [19683, 19565, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(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(19) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3124, 19716, F3, 20) (dual of [19716, 19592, 21]-code), using
- 31 times duplication [i] based on linear OOA(3124, 9858, F3, 2, 20) (dual of [(9858, 2), 19592, 21]-NRT-code), using
(105, 105+20, 2084298)-Net in Base 3 — Upper bound on s
There is no (105, 125, 2084299)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 436673 583321 062985 512943 194462 727890 303621 225909 978152 877541 > 3125 [i]