Best Known (38−6, 38, s)-Nets in Base 3
(38−6, 38, 6567)-Net over F3 — Constructive and digital
Digital (32, 38, 6567)-net over F3, using
- net defined by OOA [i] based on linear OOA(338, 6567, F3, 6, 6) (dual of [(6567, 6), 39364, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(338, 6567, F3, 5, 6) (dual of [(6567, 5), 32797, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(338, 19701, F3, 6) (dual of [19701, 19663, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(338, 19702, F3, 6) (dual of [19702, 19664, 7]-code), using
- construction X 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(31, 19, F3, 1) (dual of [19, 18, 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(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(338, 19702, F3, 6) (dual of [19702, 19664, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(338, 19701, F3, 6) (dual of [19701, 19663, 7]-code), using
- appending kth column [i] based on linear OOA(338, 6567, F3, 5, 6) (dual of [(6567, 5), 32797, 7]-NRT-code), using
(38−6, 38, 19703)-Net over F3 — Digital
Digital (32, 38, 19703)-net over F3, using
- net defined by OOA [i] based on linear OOA(338, 19703, F3, 6, 6) (dual of [(19703, 6), 118180, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(338, 19703, F3, 5, 6) (dual of [(19703, 5), 98477, 7]-NRT-code), using
- embedding of OOA with Gilbert–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
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(338, 19703, F3, 6) (dual of [19703, 19665, 7]-code), using
- appending kth column [i] based on linear OOA(338, 19703, F3, 5, 6) (dual of [(19703, 5), 98477, 7]-NRT-code), using
(38−6, 38, 1004358)-Net in Base 3 — Upper bound on s
There is no (32, 38, 1004359)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 350855 534675 193115 > 338 [i]