Best Known (121, 144, s)-Nets in Base 3
(121, 144, 1792)-Net over F3 — Constructive and digital
Digital (121, 144, 1792)-net over F3, using
- 32 times duplication [i] based on digital (119, 142, 1792)-net over F3, using
- net defined by OOA [i] based on linear OOA(3142, 1792, F3, 23, 23) (dual of [(1792, 23), 41074, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3142, 19713, F3, 23) (dual of [19713, 19571, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 19716, F3, 23) (dual of [19716, 19574, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3142, 19716, F3, 23) (dual of [19716, 19574, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3142, 19713, F3, 23) (dual of [19713, 19571, 24]-code), using
- net defined by OOA [i] based on linear OOA(3142, 1792, F3, 23, 23) (dual of [(1792, 23), 41074, 24]-NRT-code), using
(121, 144, 9859)-Net over F3 — Digital
Digital (121, 144, 9859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3144, 9859, F3, 2, 23) (dual of [(9859, 2), 19574, 24]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3142, 9858, F3, 2, 23) (dual of [(9858, 2), 19574, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3142, 19716, F3, 23) (dual of [19716, 19574, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(3142, 19716, F3, 23) (dual of [19716, 19574, 24]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3142, 9858, F3, 2, 23) (dual of [(9858, 2), 19574, 24]-NRT-code), using
(121, 144, 3913445)-Net in Base 3 — Upper bound on s
There is no (121, 144, 3913446)-net in base 3, because
- 1 times m-reduction [i] would yield (121, 143, 3913446)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 169 176428 735629 315536 478935 375103 776297 142352 417772 270057 448024 791953 > 3143 [i]