Best Known (234−28, 234, s)-Nets in Base 3
(234−28, 234, 37970)-Net over F3 — Constructive and digital
Digital (206, 234, 37970)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (189, 217, 37960)-net over F3, using
- net defined by OOA [i] based on linear OOA(3217, 37960, F3, 28, 28) (dual of [(37960, 28), 1062663, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3217, 531440, F3, 28) (dual of [531440, 531223, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3217, 531440, F3, 28) (dual of [531440, 531223, 29]-code), using
- net defined by OOA [i] based on linear OOA(3217, 37960, F3, 28, 28) (dual of [(37960, 28), 1062663, 29]-NRT-code), using
- digital (3, 17, 10)-net over F3, using
(234−28, 234, 177172)-Net over F3 — Digital
Digital (206, 234, 177172)-net over F3, using
- 31 times duplication [i] based on digital (205, 233, 177172)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 177172, F3, 3, 28) (dual of [(177172, 3), 531283, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3233, 531516, F3, 28) (dual of [531516, 531283, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 531517, F3, 28) (dual of [531517, 531284, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(316, 76, F3, 7) (dual of [76, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3233, 531517, F3, 28) (dual of [531517, 531284, 29]-code), using
- OOA 3-folding [i] based on linear OA(3233, 531516, F3, 28) (dual of [531516, 531283, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 177172, F3, 3, 28) (dual of [(177172, 3), 531283, 29]-NRT-code), using
(234−28, 234, large)-Net in Base 3 — Upper bound on s
There is no (206, 234, large)-net in base 3, because
- 26 times m-reduction [i] would yield (206, 208, large)-net in base 3, but