Best Known (230−26, 230, s)-Nets in Base 3
(230−26, 230, 122643)-Net over F3 — Constructive and digital
Digital (204, 230, 122643)-net over F3, using
- 32 times duplication [i] based on digital (202, 228, 122643)-net over F3, using
- net defined by OOA [i] based on linear OOA(3228, 122643, F3, 26, 26) (dual of [(122643, 26), 3188490, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3228, 1594359, F3, 26) (dual of [1594359, 1594131, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3228, 1594368, F3, 26) (dual of [1594368, 1594140, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3228, 1594368, F3, 26) (dual of [1594368, 1594140, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3228, 1594359, F3, 26) (dual of [1594359, 1594131, 27]-code), using
- net defined by OOA [i] based on linear OOA(3228, 122643, F3, 26, 26) (dual of [(122643, 26), 3188490, 27]-NRT-code), using
(230−26, 230, 398592)-Net over F3 — Digital
Digital (204, 230, 398592)-net over F3, using
- 32 times duplication [i] based on digital (202, 228, 398592)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3228, 398592, F3, 4, 26) (dual of [(398592, 4), 1594140, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3228, 1594368, F3, 26) (dual of [1594368, 1594140, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3222, 1594323, F3, 26) (dual of [1594323, 1594101, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3183, 1594323, F3, 22) (dual of [1594323, 1594140, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(36, 45, F3, 3) (dual of [45, 39, 4]-code or 45-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(25) ⊂ Ce(21) [i] based on
- OOA 4-folding [i] based on linear OA(3228, 1594368, F3, 26) (dual of [1594368, 1594140, 27]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3228, 398592, F3, 4, 26) (dual of [(398592, 4), 1594140, 27]-NRT-code), using
(230−26, 230, large)-Net in Base 3 — Upper bound on s
There is no (204, 230, large)-net in base 3, because
- 24 times m-reduction [i] would yield (204, 206, large)-net in base 3, but