Best Known (56, 67, s)-Nets in Base 3
(56, 67, 3939)-Net over F3 — Constructive and digital
Digital (56, 67, 3939)-net over F3, using
- net defined by OOA [i] based on linear OOA(367, 3939, F3, 11, 11) (dual of [(3939, 11), 43262, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(367, 19696, F3, 11) (dual of [19696, 19629, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(364, 19683, F3, 11) (dual of [19683, 19619, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(346, 19683, F3, 8) (dual of [19683, 19637, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(367, 19696, F3, 11) (dual of [19696, 19629, 12]-code), using
(56, 67, 9848)-Net over F3 — Digital
Digital (56, 67, 9848)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(367, 9848, F3, 2, 11) (dual of [(9848, 2), 19629, 12]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(365, 9847, F3, 2, 11) (dual of [(9847, 2), 19629, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(365, 19694, F3, 11) (dual of [19694, 19629, 12]-code), using
- construction X4 applied to C([0,10]) ⊂ C([1,9]) [i] based on
- linear OA(364, 19682, F3, 11) (dual of [19682, 19618, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(354, 19682, F3, 9) (dual of [19682, 19628, 10]-code), using the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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 C([0,10]) ⊂ C([1,9]) [i] based on
- OOA 2-folding [i] based on linear OA(365, 19694, F3, 11) (dual of [19694, 19629, 12]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(365, 9847, F3, 2, 11) (dual of [(9847, 2), 19629, 12]-NRT-code), using
(56, 67, 2587057)-Net in Base 3 — Upper bound on s
There is no (56, 67, 2587058)-net in base 3, because
- 1 times m-reduction [i] would yield (56, 66, 2587058)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 30 903191 731803 015092 263528 869885 > 366 [i]