Best Known (31−5, 31, s)-Nets in Base 3
(31−5, 31, 29529)-Net over F3 — Constructive and digital
Digital (26, 31, 29529)-net over F3, using
- net defined by OOA [i] based on linear OOA(331, 29529, F3, 5, 5) (dual of [(29529, 5), 147614, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(331, 59059, F3, 5) (dual of [59059, 59028, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(321, 59049, F3, 4) (dual of [59049, 59028, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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
- OOA 2-folding and stacking with additional row [i] based on linear OA(331, 59059, F3, 5) (dual of [59059, 59028, 6]-code), using
(31−5, 31, 53648)-Net over F3 — Digital
Digital (26, 31, 53648)-net over F3, using
- net defined by OOA [i] based on linear OOA(331, 53648, F3, 5, 5) (dual of [(53648, 5), 268209, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(331, 53648, F3, 4, 5) (dual of [(53648, 4), 214561, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(331, 53648, F3, 5) (dual of [53648, 53617, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(331, 59049, F3, 5) (dual of [59049, 59018, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(331, 53648, F3, 5) (dual of [53648, 53617, 6]-code), using
- appending kth column [i] based on linear OOA(331, 53648, F3, 4, 5) (dual of [(53648, 4), 214561, 6]-NRT-code), using
(31−5, 31, large)-Net in Base 3 — Upper bound on s
There is no (26, 31, large)-net in base 3, because
- 3 times m-reduction [i] would yield (26, 28, large)-net in base 3, but