Best Known (66, 66+8, s)-Nets in Base 3
(66, 66+8, 1195746)-Net over F3 — Constructive and digital
Digital (66, 74, 1195746)-net over F3, using
- 32 times duplication [i] based on digital (64, 72, 1195746)-net over F3, using
- net defined by OOA [i] based on linear OOA(372, 1195746, F3, 8, 8) (dual of [(1195746, 8), 9565896, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(372, 4782984, F3, 8) (dual of [4782984, 4782912, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(372, 4782985, F3, 8) (dual of [4782985, 4782913, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(371, 4782969, F3, 8) (dual of [4782969, 4782898, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(357, 4782969, F3, 7) (dual of [4782969, 4782912, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(315, 16, F3, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,3)), using
- dual of repetition code with length 16 [i]
- linear OA(31, 16, F3, 1) (dual of [16, 15, 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(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(372, 4782985, F3, 8) (dual of [4782985, 4782913, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(372, 4782984, F3, 8) (dual of [4782984, 4782912, 9]-code), using
- net defined by OOA [i] based on linear OOA(372, 1195746, F3, 8, 8) (dual of [(1195746, 8), 9565896, 9]-NRT-code), using
(66, 66+8, 2391494)-Net over F3 — Digital
Digital (66, 74, 2391494)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(374, 2391494, F3, 2, 8) (dual of [(2391494, 2), 4782914, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(374, 4782988, F3, 8) (dual of [4782988, 4782914, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(371, 4782983, F3, 8) (dual of [4782983, 4782912, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(371, 4782969, F3, 8) (dual of [4782969, 4782898, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(357, 4782969, F3, 7) (dual of [4782969, 4782912, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 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(7) ⊂ Ce(6) [i] based on
- linear OA(371, 4782985, F3, 6) (dual of [4782985, 4782914, 7]-code), using Gilbert–Varšamov bound and bm = 371 > Vbs−1(k−1) = 667 517554 975509 645021 855027 463073 [i]
- linear OA(31, 3, F3, 1) (dual of [3, 2, 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
- linear OA(371, 4782983, F3, 8) (dual of [4782983, 4782912, 9]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(374, 4782988, F3, 8) (dual of [4782988, 4782914, 9]-code), using
(66, 66+8, large)-Net in Base 3 — Upper bound on s
There is no (66, 74, large)-net in base 3, because
- 6 times m-reduction [i] would yield (66, 68, large)-net in base 3, but