Best Known (39−6, 39, s)-Nets in Base 3
(39−6, 39, 6568)-Net over F3 — Constructive and digital
Digital (33, 39, 6568)-net over F3, using
- net defined by OOA [i] based on linear OOA(339, 6568, F3, 6, 6) (dual of [(6568, 6), 39369, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(339, 6568, F3, 5, 6) (dual of [(6568, 5), 32801, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(339, 19704, F3, 6) (dual of [19704, 19665, 7]-code), using
- 1 times code embedding in larger space [i] based on linear OA(338, 19703, F3, 6) (dual of [19703, 19665, 7]-code), using
- construction X4 applied to Ce(6) ⊂ 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(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(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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(3) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(338, 19703, F3, 6) (dual of [19703, 19665, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(339, 19704, F3, 6) (dual of [19704, 19665, 7]-code), using
- appending kth column [i] based on linear OOA(339, 6568, F3, 5, 6) (dual of [(6568, 5), 32801, 7]-NRT-code), using
(39−6, 39, 19705)-Net over F3 — Digital
Digital (33, 39, 19705)-net over F3, using
- net defined by OOA [i] based on linear OOA(339, 19705, F3, 6, 6) (dual of [(19705, 6), 118191, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(339, 19705, F3, 5, 6) (dual of [(19705, 5), 98486, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 19705, F3, 6) (dual of [19705, 19666, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(338, 19703, F3, 6) (dual of [19703, 19665, 7]-code), using
- construction X4 applied to Ce(6) ⊂ 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(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(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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(3) [i] based on
- linear OA(338, 19704, F3, 5) (dual of [19704, 19666, 6]-code), using Gilbert–Varšamov bound and bm = 338 > Vbs−1(k−1) = 100450 014116 830827 [i]
- linear OA(30, 1, F3, 0) (dual of [1, 1, 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
- linear OA(338, 19703, F3, 6) (dual of [19703, 19665, 7]-code), using
- construction X with Varšamov bound [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(339, 19705, F3, 6) (dual of [19705, 19666, 7]-code), using
- appending kth column [i] based on linear OOA(339, 19705, F3, 5, 6) (dual of [(19705, 5), 98486, 7]-NRT-code), using
(39−6, 39, 1448536)-Net in Base 3 — Upper bound on s
There is no (33, 39, 1448537)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 052562 895914 035771 > 339 [i]