Best Known (226−27, 226, s)-Nets in Base 3
(226−27, 226, 40884)-Net over F3 — Constructive and digital
Digital (199, 226, 40884)-net over F3, using
- net defined by OOA [i] based on linear OOA(3226, 40884, F3, 27, 27) (dual of [(40884, 27), 1103642, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3226, 531493, F3, 27) (dual of [531493, 531267, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3226, 531498, F3, 27) (dual of [531498, 531272, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [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(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 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(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3226, 531498, F3, 27) (dual of [531498, 531272, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3226, 531493, F3, 27) (dual of [531493, 531267, 28]-code), using
(226−27, 226, 177166)-Net over F3 — Digital
Digital (199, 226, 177166)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3226, 177166, F3, 3, 27) (dual of [(177166, 3), 531272, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3226, 531498, F3, 27) (dual of [531498, 531272, 28]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [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(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(39, 57, F3, 4) (dual of [57, 48, 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(21) [i] based on
- OOA 3-folding [i] based on linear OA(3226, 531498, F3, 27) (dual of [531498, 531272, 28]-code), using
(226−27, 226, large)-Net in Base 3 — Upper bound on s
There is no (199, 226, large)-net in base 3, because
- 25 times m-reduction [i] would yield (199, 201, large)-net in base 3, but