Best Known (37, 37+7, s)-Nets in Base 3
(37, 37+7, 19687)-Net over F3 — Constructive and digital
Digital (37, 44, 19687)-net over F3, using
- 31 times duplication [i] based on digital (36, 43, 19687)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 19687, F3, 7, 7) (dual of [(19687, 7), 137766, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(343, 59062, F3, 7) (dual of [59062, 59019, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(4) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(343, 59062, F3, 7) (dual of [59062, 59019, 8]-code), using
- net defined by OOA [i] based on linear OOA(343, 19687, F3, 7, 7) (dual of [(19687, 7), 137766, 8]-NRT-code), using
(37, 37+7, 29532)-Net over F3 — Digital
Digital (37, 44, 29532)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(344, 29532, F3, 2, 7) (dual of [(29532, 2), 59020, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(344, 59064, F3, 7) (dual of [59064, 59020, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(4) [i] based on
- linear OA(342, 59062, F3, 5) (dual of [59062, 59020, 6]-code), using Gilbert–Varšamov bound and bm = 342 > Vbs−1(k−1) = 8 111151 471710 321043 [i]
- linear OA(31, 2, F3, 1) (dual of [2, 1, 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 (see above)
- linear OA(342, 59061, F3, 7) (dual of [59061, 59019, 8]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(344, 59064, F3, 7) (dual of [59064, 59020, 8]-code), using
(37, 37+7, 6267459)-Net in Base 3 — Upper bound on s
There is no (37, 44, 6267460)-net in base 3, because
- 1 times m-reduction [i] would yield (37, 43, 6267460)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 328 256979 530919 020081 > 343 [i]