Best Known (230−28, 230, s)-Nets in Base 3
(230−28, 230, 37964)-Net over F3 — Constructive and digital
Digital (202, 230, 37964)-net over F3, using
- 32 times duplication [i] based on digital (200, 228, 37964)-net over F3, using
- net defined by OOA [i] based on linear OOA(3228, 37964, F3, 28, 28) (dual of [(37964, 28), 1062764, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3228, 531496, F3, 28) (dual of [531496, 531268, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 531500, F3, 28) (dual of [531500, 531272, 29]-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(311, 59, F3, 5) (dual of [59, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 531500, F3, 28) (dual of [531500, 531272, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3228, 531496, F3, 28) (dual of [531496, 531268, 29]-code), using
- net defined by OOA [i] based on linear OOA(3228, 37964, F3, 28, 28) (dual of [(37964, 28), 1062764, 29]-NRT-code), using
(230−28, 230, 159603)-Net over F3 — Digital
Digital (202, 230, 159603)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3230, 159603, F3, 3, 28) (dual of [(159603, 3), 478579, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3230, 177167, F3, 3, 28) (dual of [(177167, 3), 531271, 29]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3229, 177167, F3, 3, 28) (dual of [(177167, 3), 531272, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3229, 531501, F3, 28) (dual of [531501, 531272, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3228, 531500, F3, 28) (dual of [531500, 531272, 29]-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(311, 59, F3, 5) (dual of [59, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3228, 531500, F3, 28) (dual of [531500, 531272, 29]-code), using
- OOA 3-folding [i] based on linear OA(3229, 531501, F3, 28) (dual of [531501, 531272, 29]-code), using
- 31 times duplication [i] based on linear OOA(3229, 177167, F3, 3, 28) (dual of [(177167, 3), 531272, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3230, 177167, F3, 3, 28) (dual of [(177167, 3), 531271, 29]-NRT-code), using
(230−28, 230, large)-Net in Base 3 — Upper bound on s
There is no (202, 230, large)-net in base 3, because
- 26 times m-reduction [i] would yield (202, 204, large)-net in base 3, but