Best Known (230−29, 230, s)-Nets in Base 3
(230−29, 230, 37960)-Net over F3 — Constructive and digital
Digital (201, 230, 37960)-net over F3, using
- 31 times duplication [i] based on digital (200, 229, 37960)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 37960, F3, 29, 29) (dual of [(37960, 29), 1100611, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 14-folding and stacking with additional row [i] based on linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using
- net defined by OOA [i] based on linear OOA(3229, 37960, F3, 29, 29) (dual of [(37960, 29), 1100611, 30]-NRT-code), using
(230−29, 230, 132863)-Net over F3 — Digital
Digital (201, 230, 132863)-net over F3, using
- 31 times duplication [i] based on digital (200, 229, 132863)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3229, 132863, F3, 4, 29) (dual of [(132863, 4), 531223, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3229, 531452, F3, 29) (dual of [531452, 531223, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(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(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 531453, F3, 29) (dual of [531453, 531224, 30]-code), using
- OOA 4-folding [i] based on linear OA(3229, 531452, F3, 29) (dual of [531452, 531223, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3229, 132863, F3, 4, 29) (dual of [(132863, 4), 531223, 30]-NRT-code), using
(230−29, 230, large)-Net in Base 3 — Upper bound on s
There is no (201, 230, large)-net in base 3, because
- 27 times m-reduction [i] would yield (201, 203, large)-net in base 3, but