Best Known (33, 40, s)-Nets in Base 3
(33, 40, 6565)-Net over F3 — Constructive and digital
Digital (33, 40, 6565)-net over F3, using
- net defined by OOA [i] based on linear OOA(340, 6565, F3, 7, 7) (dual of [(6565, 7), 45915, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(340, 19696, F3, 7) (dual of [19696, 19656, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(340, 19697, F3, 7) (dual of [19697, 19657, 8]-code), using
- construction XX applied to Ce(6) ⊂ Ce(4) ⊂ Ce(3) [i] based on
- linear OA(337, 19683, F3, 7) (dual of [19683, 19646, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(328, 19683, F3, 5) (dual of [19683, 19655, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(319, 19683, F3, 4) (dual of [19683, 19664, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- linear OA(30, 2, F3, 0) (dual of [2, 2, 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 XX applied to Ce(6) ⊂ Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(340, 19697, F3, 7) (dual of [19697, 19657, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(340, 19696, F3, 7) (dual of [19696, 19656, 8]-code), using
(33, 40, 9848)-Net over F3 — Digital
Digital (33, 40, 9848)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(340, 9848, F3, 2, 7) (dual of [(9848, 2), 19656, 8]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(338, 9847, F3, 2, 7) (dual of [(9847, 2), 19656, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(338, 19694, F3, 7) (dual of [19694, 19656, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(337, 19683, F3, 7) (dual of [19683, 19646, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(328, 19683, F3, 5) (dual of [19683, 19655, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(310, 11, F3, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
- dual of repetition code with length 11 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(338, 19694, F3, 7) (dual of [19694, 19656, 8]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(338, 9847, F3, 2, 7) (dual of [(9847, 2), 19656, 8]-NRT-code), using
(33, 40, 1448536)-Net in Base 3 — Upper bound on s
There is no (33, 40, 1448537)-net in base 3, because
- 1 times m-reduction [i] would yield (33, 39, 1448537)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 052562 895914 035771 > 339 [i]