Best Known (80, 80+9, s)-Nets in Base 3
(80, 80+9, 1195750)-Net over F3 — Constructive and digital
Digital (80, 89, 1195750)-net over F3, using
- 32 times duplication [i] based on digital (78, 87, 1195750)-net over F3, using
- net defined by OOA [i] based on linear OOA(387, 1195750, F3, 9, 9) (dual of [(1195750, 9), 10761663, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(387, 4783001, F3, 9) (dual of [4783001, 4782914, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(385, 4782970, F3, 9) (dual of [4782970, 4782885, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(387, 4783001, F3, 9) (dual of [4783001, 4782914, 10]-code), using
- net defined by OOA [i] based on linear OOA(387, 1195750, F3, 9, 9) (dual of [(1195750, 9), 10761663, 10]-NRT-code), using
(80, 80+9, 2391501)-Net over F3 — Digital
Digital (80, 89, 2391501)-net over F3, using
- 31 times duplication [i] based on digital (79, 88, 2391501)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(388, 2391501, F3, 2, 9) (dual of [(2391501, 2), 4782914, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(386, 2391500, F3, 2, 9) (dual of [(2391500, 2), 4782914, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(385, 4782970, F3, 9) (dual of [4782970, 4782885, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(357, 4782970, F3, 7) (dual of [4782970, 4782913, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(386, 4783000, F3, 9) (dual of [4783000, 4782914, 10]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(386, 2391500, F3, 2, 9) (dual of [(2391500, 2), 4782914, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(388, 2391501, F3, 2, 9) (dual of [(2391501, 2), 4782914, 10]-NRT-code), using
(80, 80+9, large)-Net in Base 3 — Upper bound on s
There is no (80, 89, large)-net in base 3, because
- 7 times m-reduction [i] would yield (80, 82, large)-net in base 3, but