Best Known (197, 197+28, s)-Nets in Base 3
(197, 197+28, 37963)-Net over F3 — Constructive and digital
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
(197, 197+28, 132870)-Net over F3 — Digital
Digital (197, 225, 132870)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 132870, F3, 4, 28) (dual of [(132870, 4), 531255, 29]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3225, 531480, F3, 28) (dual of [531480, 531255, 29]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3225, 531482, F3, 28) (dual of [531482, 531257, 29]-code), using
- OOA 4-folding [i] based on linear OA(3225, 531480, F3, 28) (dual of [531480, 531255, 29]-code), using
(197, 197+28, large)-Net in Base 3 — Upper bound on s
There is no (197, 225, large)-net in base 3, because
- 26 times m-reduction [i] would yield (197, 199, large)-net in base 3, but