Best Known (126, 126+17, s)-Nets in Base 3
(126, 126+17, 66439)-Net over F3 — Constructive and digital
Digital (126, 143, 66439)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (116, 133, 66431)-net over F3, using
- net defined by OOA [i] based on linear OOA(3133, 66431, F3, 17, 17) (dual of [(66431, 17), 1129194, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3133, 531449, F3, 17) (dual of [531449, 531316, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 531453, F3, 17) (dual of [531453, 531320, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3133, 531449, F3, 17) (dual of [531449, 531316, 18]-code), using
- net defined by OOA [i] based on linear OOA(3133, 66431, F3, 17, 17) (dual of [(66431, 17), 1129194, 18]-NRT-code), using
- digital (2, 10, 8)-net over F3, using
(126, 126+17, 193059)-Net over F3 — Digital
Digital (126, 143, 193059)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3143, 193059, F3, 2, 17) (dual of [(193059, 2), 385975, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3143, 265743, F3, 2, 17) (dual of [(265743, 2), 531343, 18]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3142, 265743, F3, 2, 17) (dual of [(265743, 2), 531344, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3140, 265742, F3, 2, 17) (dual of [(265742, 2), 531344, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3140, 531484, F3, 17) (dual of [531484, 531344, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3133, 531441, F3, 17) (dual of [531441, 531308, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3139, 531483, F3, 17) (dual of [531483, 531344, 18]-code), using
- OOA 2-folding [i] based on linear OA(3140, 531484, F3, 17) (dual of [531484, 531344, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3140, 265742, F3, 2, 17) (dual of [(265742, 2), 531344, 18]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3142, 265743, F3, 2, 17) (dual of [(265743, 2), 531344, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3143, 265743, F3, 2, 17) (dual of [(265743, 2), 531343, 18]-NRT-code), using
(126, 126+17, large)-Net in Base 3 — Upper bound on s
There is no (126, 143, large)-net in base 3, because
- 15 times m-reduction [i] would yield (126, 128, large)-net in base 3, but