Best Known (76−9, 76, s)-Nets in Base 3
(76−9, 76, 132867)-Net over F3 — Constructive and digital
Digital (67, 76, 132867)-net over F3, using
- 31 times duplication [i] based on digital (66, 75, 132867)-net over F3, using
- net defined by OOA [i] based on linear OOA(375, 132867, F3, 9, 9) (dual of [(132867, 9), 1195728, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(375, 531469, F3, 9) (dual of [531469, 531394, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(374, 531468, F3, 9) (dual of [531468, 531394, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(373, 531442, F3, 9) (dual of [531442, 531369, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(349, 531442, F3, 7) (dual of [531442, 531393, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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(374, 531468, F3, 9) (dual of [531468, 531394, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(375, 531469, F3, 9) (dual of [531469, 531394, 10]-code), using
- net defined by OOA [i] based on linear OOA(375, 132867, F3, 9, 9) (dual of [(132867, 9), 1195728, 10]-NRT-code), using
(76−9, 76, 265735)-Net over F3 — Digital
Digital (67, 76, 265735)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(376, 265735, F3, 2, 9) (dual of [(265735, 2), 531394, 10]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(374, 265734, F3, 2, 9) (dual of [(265734, 2), 531394, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(374, 531468, F3, 9) (dual of [531468, 531394, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(373, 531442, F3, 9) (dual of [531442, 531369, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(349, 531442, F3, 7) (dual of [531442, 531393, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 324−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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(374, 531468, F3, 9) (dual of [531468, 531394, 10]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(374, 265734, F3, 2, 9) (dual of [(265734, 2), 531394, 10]-NRT-code), using
(76−9, 76, large)-Net in Base 3 — Upper bound on s
There is no (67, 76, large)-net in base 3, because
- 7 times m-reduction [i] would yield (67, 69, large)-net in base 3, but