Best Known (209, 209+32, s)-Nets in Base 3
(209, 209+32, 11074)-Net over F3 — Constructive and digital
Digital (209, 241, 11074)-net over F3, using
- 33 times duplication [i] based on digital (206, 238, 11074)-net over F3, using
- net defined by OOA [i] based on linear OOA(3238, 11074, F3, 32, 32) (dual of [(11074, 32), 354130, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3238, 177184, F3, 32) (dual of [177184, 176946, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 177186, F3, 32) (dual of [177186, 176948, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-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(31) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 177186, F3, 32) (dual of [177186, 176948, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3238, 177184, F3, 32) (dual of [177184, 176946, 33]-code), using
- net defined by OOA [i] based on linear OOA(3238, 11074, F3, 32, 32) (dual of [(11074, 32), 354130, 33]-NRT-code), using
(209, 209+32, 59063)-Net over F3 — Digital
Digital (209, 241, 59063)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3241, 59063, F3, 3, 32) (dual of [(59063, 3), 176948, 33]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3238, 59062, F3, 3, 32) (dual of [(59062, 3), 176948, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3238, 177186, F3, 32) (dual of [177186, 176948, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-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(31) ⊂ Ce(27) [i] based on
- OOA 3-folding [i] based on linear OA(3238, 177186, F3, 32) (dual of [177186, 176948, 33]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3238, 59062, F3, 3, 32) (dual of [(59062, 3), 176948, 33]-NRT-code), using
(209, 209+32, large)-Net in Base 3 — Upper bound on s
There is no (209, 241, large)-net in base 3, because
- 30 times m-reduction [i] would yield (209, 211, large)-net in base 3, but