Best Known (224−27, 224, s)-Nets in Base 3
(224−27, 224, 40883)-Net over F3 — Constructive and digital
Digital (197, 224, 40883)-net over F3, using
- 31 times duplication [i] based on digital (196, 223, 40883)-net over F3, using
- net defined by OOA [i] based on linear OOA(3223, 40883, F3, 27, 27) (dual of [(40883, 27), 1103618, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3223, 531480, F3, 27) (dual of [531480, 531257, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3223, 531483, F3, 27) (dual of [531483, 531260, 28]-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(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(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3223, 531483, F3, 27) (dual of [531483, 531260, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3223, 531480, F3, 27) (dual of [531480, 531257, 28]-code), using
- net defined by OOA [i] based on linear OOA(3223, 40883, F3, 27, 27) (dual of [(40883, 27), 1103618, 28]-NRT-code), using
(224−27, 224, 177161)-Net over F3 — Digital
Digital (197, 224, 177161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3224, 177161, F3, 3, 27) (dual of [(177161, 3), 531259, 28]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3223, 177161, F3, 3, 27) (dual of [(177161, 3), 531260, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3223, 531483, F3, 27) (dual of [531483, 531260, 28]-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(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(27) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(3223, 531483, F3, 27) (dual of [531483, 531260, 28]-code), using
- 31 times duplication [i] based on linear OOA(3223, 177161, F3, 3, 27) (dual of [(177161, 3), 531260, 28]-NRT-code), using
(224−27, 224, large)-Net in Base 3 — Upper bound on s
There is no (197, 224, large)-net in base 3, because
- 25 times m-reduction [i] would yield (197, 199, large)-net in base 3, but