Best Known (199, 199+28, s)-Nets in Base 3
(199, 199+28, 37963)-Net over F3 — Constructive and digital
Digital (199, 227, 37963)-net over F3, using
- 32 times duplication [i] based on digital (197, 225, 37963)-net over F3, using
- net defined by OOA [i] based on linear OOA(3225, 37963, F3, 28, 28) (dual of [(37963, 28), 1062739, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3225, 531482, F3, 28) (dual of [531482, 531257, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(22) [i] based on
- OA 14-folding and stacking [i] based on linear OA(3225, 531482, F3, 28) (dual of [531482, 531257, 29]-code), using
- net defined by OOA [i] based on linear OOA(3225, 37963, F3, 28, 28) (dual of [(37963, 28), 1062739, 29]-NRT-code), using
(199, 199+28, 139121)-Net over F3 — Digital
Digital (199, 227, 139121)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 139121, F3, 3, 28) (dual of [(139121, 3), 417136, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3227, 177162, F3, 3, 28) (dual of [(177162, 3), 531259, 29]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3226, 177162, F3, 3, 28) (dual of [(177162, 3), 531260, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3226, 531486, F3, 28) (dual of [531486, 531260, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(3226, 531486, F3, 28) (dual of [531486, 531260, 29]-code), using
- 31 times duplication [i] based on linear OOA(3226, 177162, F3, 3, 28) (dual of [(177162, 3), 531260, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3227, 177162, F3, 3, 28) (dual of [(177162, 3), 531259, 29]-NRT-code), using
(199, 199+28, large)-Net in Base 3 — Upper bound on s
There is no (199, 227, large)-net in base 3, because
- 26 times m-reduction [i] would yield (199, 201, large)-net in base 3, but