Best Known (38−5, 38, s)-Nets in Base 3
(38−5, 38, 265727)-Net over F3 — Constructive and digital
Digital (33, 38, 265727)-net over F3, using
- net defined by OOA [i] based on linear OOA(338, 265727, F3, 5, 5) (dual of [(265727, 5), 1328597, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(338, 531455, F3, 5) (dual of [531455, 531417, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(338, 531455, F3, 5) (dual of [531455, 531417, 6]-code), using
(38−5, 38, 531455)-Net over F3 — Digital
Digital (33, 38, 531455)-net over F3, using
- net defined by OOA [i] based on linear OOA(338, 531455, F3, 5, 5) (dual of [(531455, 5), 2657237, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(338, 531455, F3, 4, 5) (dual of [(531455, 4), 2125782, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(338, 531455, F3, 5) (dual of [531455, 531417, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(4) ⊂ Ce(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(338, 531455, F3, 5) (dual of [531455, 531417, 6]-code), using
- appending kth column [i] based on linear OOA(338, 531455, F3, 4, 5) (dual of [(531455, 4), 2125782, 6]-NRT-code), using
(38−5, 38, large)-Net in Base 3 — Upper bound on s
There is no (33, 38, large)-net in base 3, because
- 3 times m-reduction [i] would yield (33, 35, large)-net in base 3, but