Best Known (65, 65+8, s)-Nets in Base 3
(65, 65+8, 1195746)-Net over F3 — Constructive and digital
Digital (65, 73, 1195746)-net over F3, using
- 31 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
(65, 65+8, 2391493)-Net over F3 — Digital
Digital (65, 73, 2391493)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(373, 2391493, F3, 2, 8) (dual of [(2391493, 2), 4782913, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(373, 4782986, F3, 8) (dual of [4782986, 4782913, 9]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(372, 4782985, F3, 8) (dual of [4782985, 4782913, 9]-code), using
- OOA 2-folding [i] based on linear OA(373, 4782986, F3, 8) (dual of [4782986, 4782913, 9]-code), using
(65, 65+8, large)-Net in Base 3 — Upper bound on s
There is no (65, 73, large)-net in base 3, because
- 6 times m-reduction [i] would yield (65, 67, large)-net in base 3, but