Best Known (159, 159+22, s)-Nets in Base 3
(159, 159+22, 48319)-Net over F3 — Constructive and digital
Digital (159, 181, 48319)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (147, 169, 48312)-net over F3, using
- net defined by OOA [i] based on linear OOA(3169, 48312, F3, 22, 22) (dual of [(48312, 22), 1062695, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(3169, 531432, F3, 22) (dual of [531432, 531263, 23]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(3169, 531432, F3, 22) (dual of [531432, 531263, 23]-code), using
- net defined by OOA [i] based on linear OOA(3169, 48312, F3, 22, 22) (dual of [(48312, 22), 1062695, 23]-NRT-code), using
- digital (1, 12, 7)-net over F3, using
(159, 159+22, 177167)-Net over F3 — Digital
Digital (159, 181, 177167)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3181, 177167, F3, 3, 22) (dual of [(177167, 3), 531320, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3181, 531501, F3, 22) (dual of [531501, 531320, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3180, 531500, F3, 22) (dual of [531500, 531320, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- 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(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(21) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3180, 531500, F3, 22) (dual of [531500, 531320, 23]-code), using
- OOA 3-folding [i] based on linear OA(3181, 531501, F3, 22) (dual of [531501, 531320, 23]-code), using
(159, 159+22, large)-Net in Base 3 — Upper bound on s
There is no (159, 181, large)-net in base 3, because
- 20 times m-reduction [i] would yield (159, 161, large)-net in base 3, but