Best Known (80, 90, s)-Nets in Base 3
(80, 90, 956600)-Net over F3 — Constructive and digital
Digital (80, 90, 956600)-net over F3, using
- 31 times duplication [i] based on digital (79, 89, 956600)-net over F3, using
- net defined by OOA [i] based on linear OOA(389, 956600, F3, 10, 10) (dual of [(956600, 10), 9565911, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(389, 4783000, F3, 10) (dual of [4783000, 4782911, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 4783001, F3, 10) (dual of [4783001, 4782912, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(34, 32, F3, 2) (dual of [32, 28, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(389, 4783001, F3, 10) (dual of [4783001, 4782912, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(389, 4783000, F3, 10) (dual of [4783000, 4782911, 11]-code), using
- net defined by OOA [i] based on linear OOA(389, 956600, F3, 10, 10) (dual of [(956600, 10), 9565911, 11]-NRT-code), using
(80, 90, 1682608)-Net over F3 — Digital
Digital (80, 90, 1682608)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(390, 1682608, F3, 2, 10) (dual of [(1682608, 2), 3365126, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(390, 2391501, F3, 2, 10) (dual of [(2391501, 2), 4782912, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(390, 4783002, F3, 10) (dual of [4783002, 4782912, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(389, 4783001, F3, 10) (dual of [4783001, 4782912, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(34, 32, F3, 2) (dual of [32, 28, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(389, 4783001, F3, 10) (dual of [4783001, 4782912, 11]-code), using
- OOA 2-folding [i] based on linear OA(390, 4783002, F3, 10) (dual of [4783002, 4782912, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(390, 2391501, F3, 2, 10) (dual of [(2391501, 2), 4782912, 11]-NRT-code), using
(80, 90, large)-Net in Base 3 — Upper bound on s
There is no (80, 90, large)-net in base 3, because
- 8 times m-reduction [i] would yield (80, 82, large)-net in base 3, but