Best Known (27, 27+5, s)-Nets in Base 3
(27, 27+5, 29530)-Net over F3 — Constructive and digital
Digital (27, 32, 29530)-net over F3, using
- net defined by OOA [i] based on linear OOA(332, 29530, F3, 5, 5) (dual of [(29530, 5), 147618, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(332, 59061, F3, 5) (dual of [59061, 59029, 6]-code), using
- construction X4 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(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(332, 59061, F3, 5) (dual of [59061, 59029, 6]-code), using
(27, 27+5, 59061)-Net over F3 — Digital
Digital (27, 32, 59061)-net over F3, using
- net defined by OOA [i] based on linear OOA(332, 59061, F3, 5, 5) (dual of [(59061, 5), 295273, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(332, 59061, F3, 4, 5) (dual of [(59061, 4), 236212, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 59061, F3, 5) (dual of [59061, 59029, 6]-code), using
- construction X4 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(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(332, 59061, F3, 5) (dual of [59061, 59029, 6]-code), using
- appending kth column [i] based on linear OOA(332, 59061, F3, 4, 5) (dual of [(59061, 4), 236212, 6]-NRT-code), using
(27, 27+5, large)-Net in Base 3 — Upper bound on s
There is no (27, 32, large)-net in base 3, because
- 3 times m-reduction [i] would yield (27, 29, large)-net in base 3, but