Best Known (77, 77+12, s)-Nets in Base 3
(77, 77+12, 29526)-Net over F3 — Constructive and digital
Digital (77, 89, 29526)-net over F3, using
- net defined by OOA [i] based on linear OOA(389, 29526, F3, 12, 12) (dual of [(29526, 12), 354223, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(389, 177156, F3, 12) (dual of [177156, 177067, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 177159, F3, 12) (dual of [177159, 177070, 13]-code), using
- construction X4 applied to C([1,12]) ⊂ C([1,10]) [i] based on
- linear OA(388, 177146, F3, 12) (dual of [177146, 177058, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(377, 177146, F3, 10) (dual of [177146, 177069, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 C([1,12]) ⊂ C([1,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(389, 177159, F3, 12) (dual of [177159, 177070, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(389, 177156, F3, 12) (dual of [177156, 177067, 13]-code), using
(77, 77+12, 84889)-Net over F3 — Digital
Digital (77, 89, 84889)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(389, 84889, F3, 2, 12) (dual of [(84889, 2), 169689, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(389, 88579, F3, 2, 12) (dual of [(88579, 2), 177069, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(389, 177158, F3, 12) (dual of [177158, 177069, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 177159, F3, 12) (dual of [177159, 177070, 13]-code), using
- construction X4 applied to C([1,12]) ⊂ C([1,10]) [i] based on
- linear OA(388, 177146, F3, 12) (dual of [177146, 177058, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(377, 177146, F3, 10) (dual of [177146, 177069, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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 C([1,12]) ⊂ C([1,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(389, 177159, F3, 12) (dual of [177159, 177070, 13]-code), using
- OOA 2-folding [i] based on linear OA(389, 177158, F3, 12) (dual of [177158, 177069, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(389, 88579, F3, 2, 12) (dual of [(88579, 2), 177069, 13]-NRT-code), using
(77, 77+12, large)-Net in Base 3 — Upper bound on s
There is no (77, 89, large)-net in base 3, because
- 10 times m-reduction [i] would yield (77, 79, large)-net in base 3, but