Best Known (34−5, 34, s)-Nets in Base 3
(34−5, 34, 88578)-Net over F3 — Constructive and digital
Digital (29, 34, 88578)-net over F3, using
- net defined by OOA [i] based on linear OOA(334, 88578, F3, 5, 5) (dual of [(88578, 5), 442856, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(334, 177157, F3, 5) (dual of [177157, 177123, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(334, 177158, F3, 5) (dual of [177158, 177124, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(323, 177147, F3, 4) (dual of [177147, 177124, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(334, 177158, F3, 5) (dual of [177158, 177124, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(334, 177157, F3, 5) (dual of [177157, 177123, 6]-code), using
(34−5, 34, 160947)-Net over F3 — Digital
Digital (29, 34, 160947)-net over F3, using
- net defined by OOA [i] based on linear OOA(334, 160947, F3, 5, 5) (dual of [(160947, 5), 804701, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(334, 160947, F3, 4, 5) (dual of [(160947, 4), 643754, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(334, 160947, F3, 5) (dual of [160947, 160913, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(334, 177147, F3, 5) (dual of [177147, 177113, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(334, 160947, F3, 5) (dual of [160947, 160913, 6]-code), using
- appending kth column [i] based on linear OOA(334, 160947, F3, 4, 5) (dual of [(160947, 4), 643754, 6]-NRT-code), using
(34−5, 34, large)-Net in Base 3 — Upper bound on s
There is no (29, 34, large)-net in base 3, because
- 3 times m-reduction [i] would yield (29, 31, large)-net in base 3, but