Best Known (215, 215+29, s)-Nets in Base 3
(215, 215+29, 37967)-Net over F3 — Constructive and digital
Digital (215, 244, 37967)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (200, 229, 37960)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 37960, F3, 29, 29) (dual of [(37960, 29), 1100611, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [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]
- OOA 14-folding and stacking with additional row [i] based on linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using
- net defined by OOA [i] based on linear OOA(3229, 37960, F3, 29, 29) (dual of [(37960, 29), 1100611, 30]-NRT-code), using
- digital (1, 15, 7)-net over F3, using
(215, 215+29, 177172)-Net over F3 — Digital
Digital (215, 244, 177172)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3244, 177172, F3, 3, 29) (dual of [(177172, 3), 531272, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3244, 531516, F3, 29) (dual of [531516, 531272, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [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(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(315, 75, F3, 6) (dual of [75, 60, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(315, 85, F3, 6) (dual of [85, 70, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(3244, 531516, F3, 29) (dual of [531516, 531272, 30]-code), using
(215, 215+29, large)-Net in Base 3 — Upper bound on s
There is no (215, 244, large)-net in base 3, because
- 27 times m-reduction [i] would yield (215, 217, large)-net in base 3, but