Best Known (123, 123+17, s)-Nets in Base 3
(123, 123+17, 66435)-Net over F3 — Constructive and digital
Digital (123, 140, 66435)-net over F3, using
- 31 times duplication [i] based on digital (122, 139, 66435)-net over F3, using
- net defined by OOA [i] based on linear OOA(3139, 66435, F3, 17, 17) (dual of [(66435, 17), 1129256, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3139, 531481, F3, 17) (dual of [531481, 531342, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3139, 531481, F3, 17) (dual of [531481, 531342, 18]-code), using
- net defined by OOA [i] based on linear OOA(3139, 66435, F3, 17, 17) (dual of [(66435, 17), 1129256, 18]-NRT-code), using
(123, 123+17, 177161)-Net over F3 — Digital
Digital (123, 140, 177161)-net over F3, using
- 31 times duplication [i] based on digital (122, 139, 177161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3139, 177161, F3, 3, 17) (dual of [(177161, 3), 531344, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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(16) ⊂ Ce(12) [i] based on
- OOA 3-folding [i] based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3139, 177161, F3, 3, 17) (dual of [(177161, 3), 531344, 18]-NRT-code), using
(123, 123+17, large)-Net in Base 3 — Upper bound on s
There is no (123, 140, large)-net in base 3, because
- 15 times m-reduction [i] would yield (123, 125, large)-net in base 3, but