Best Known (205, 205+27, s)-Nets in Base 3
(205, 205+27, 40889)-Net over F3 — Constructive and digital
Digital (205, 232, 40889)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 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, 216, 40879)-net over F3, using
- net defined by OOA [i] based on linear OOA(3216, 40879, F3, 27, 27) (dual of [(40879, 27), 1103517, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3216, 531428, F3, 27) (dual of [531428, 531212, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 531440, F3, 27) (dual of [531440, 531224, 28]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 531440, F3, 27) (dual of [531440, 531224, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3216, 531428, F3, 27) (dual of [531428, 531212, 28]-code), using
- net defined by OOA [i] based on linear OOA(3216, 40879, F3, 27, 27) (dual of [(40879, 27), 1103517, 28]-NRT-code), using
- digital (3, 16, 10)-net over F3, using
(205, 205+27, 183101)-Net over F3 — Digital
Digital (205, 232, 183101)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 183101, F3, 2, 27) (dual of [(183101, 2), 365970, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 265758, F3, 2, 27) (dual of [(265758, 2), 531284, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3232, 531516, F3, 27) (dual of [531516, 531284, 28]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(3233, 531517, F3, 28) (dual of [531517, 531284, 29]-code), using
- OOA 2-folding [i] based on linear OA(3232, 531516, F3, 27) (dual of [531516, 531284, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 265758, F3, 2, 27) (dual of [(265758, 2), 531284, 28]-NRT-code), using
(205, 205+27, large)-Net in Base 3 — Upper bound on s
There is no (205, 232, large)-net in base 3, because
- 25 times m-reduction [i] would yield (205, 207, large)-net in base 3, but