Best Known (210, 210+29, s)-Nets in Base 3
(210, 210+29, 37963)-Net over F3 — Constructive and digital
Digital (210, 239, 37963)-net over F3, using
- 34 times duplication [i] based on digital (206, 235, 37963)-net over F3, using
- net defined by OOA [i] based on linear OOA(3235, 37963, F3, 29, 29) (dual of [(37963, 29), 1100692, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3235, 531483, F3, 29) (dual of [531483, 531248, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(3235, 531483, F3, 29) (dual of [531483, 531248, 30]-code), using
- net defined by OOA [i] based on linear OOA(3235, 37963, F3, 29, 29) (dual of [(37963, 29), 1100692, 30]-NRT-code), using
(210, 210+29, 162389)-Net over F3 — Digital
Digital (210, 239, 162389)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3239, 162389, F3, 3, 29) (dual of [(162389, 3), 486928, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 177162, F3, 3, 29) (dual of [(177162, 3), 531247, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3238, 177162, F3, 3, 29) (dual of [(177162, 3), 531248, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3235, 177161, F3, 3, 29) (dual of [(177161, 3), 531248, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3235, 531483, F3, 29) (dual of [531483, 531248, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- OOA 3-folding [i] based on linear OA(3235, 531483, F3, 29) (dual of [531483, 531248, 30]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3235, 177161, F3, 3, 29) (dual of [(177161, 3), 531248, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3238, 177162, F3, 3, 29) (dual of [(177162, 3), 531248, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3239, 177162, F3, 3, 29) (dual of [(177162, 3), 531247, 30]-NRT-code), using
(210, 210+29, large)-Net in Base 3 — Upper bound on s
There is no (210, 239, large)-net in base 3, because
- 27 times m-reduction [i] would yield (210, 212, large)-net in base 3, but