Best Known (190, 190+27, s)-Nets in Base 3
(190, 190+27, 40880)-Net over F3 — Constructive and digital
Digital (190, 217, 40880)-net over F3, using
- net defined by OOA [i] based on linear OOA(3217, 40880, F3, 27, 27) (dual of [(40880, 27), 1103543, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3217, 531441, F3, 27) (dual of [531441, 531224, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 531442, F3, 27) (dual of [531442, 531225, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3217, 531442, F3, 27) (dual of [531442, 531225, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3217, 531441, F3, 27) (dual of [531441, 531224, 28]-code), using
(190, 190+27, 132863)-Net over F3 — Digital
Digital (190, 217, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3217, 132863, F3, 4, 27) (dual of [(132863, 4), 531235, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3217, 531452, F3, 27) (dual of [531452, 531235, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 531453, F3, 27) (dual of [531453, 531236, 28]-code), using
- 1 times truncation [i] based on linear OA(3218, 531454, F3, 28) (dual of [531454, 531236, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [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(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- 1 times truncation [i] based on linear OA(3218, 531454, F3, 28) (dual of [531454, 531236, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 531453, F3, 27) (dual of [531453, 531236, 28]-code), using
- OOA 4-folding [i] based on linear OA(3217, 531452, F3, 27) (dual of [531452, 531235, 28]-code), using
(190, 190+27, large)-Net in Base 3 — Upper bound on s
There is no (190, 217, large)-net in base 3, because
- 25 times m-reduction [i] would yield (190, 192, large)-net in base 3, but