Best Known (89, 89+12, s)-Nets in Base 3
(89, 89+12, 88578)-Net over F3 — Constructive and digital
Digital (89, 101, 88578)-net over F3, using
- t-expansion [i] based on digital (88, 101, 88578)-net over F3, using
- net defined by OOA [i] based on linear OOA(3101, 88578, F3, 13, 13) (dual of [(88578, 13), 1151413, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3101, 531469, F3, 13) (dual of [531469, 531368, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(3101, 531469, F3, 13) (dual of [531469, 531368, 14]-code), using
- net defined by OOA [i] based on linear OOA(3101, 88578, F3, 13, 13) (dual of [(88578, 13), 1151413, 14]-NRT-code), using
(89, 89+12, 265735)-Net over F3 — Digital
Digital (89, 101, 265735)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3101, 265735, F3, 2, 12) (dual of [(265735, 2), 531369, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(399, 265734, F3, 2, 12) (dual of [(265734, 2), 531369, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(399, 531468, F3, 12) (dual of [531468, 531369, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(398, 531467, F3, 12) (dual of [531467, 531369, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(397, 531441, F3, 13) (dual of [531441, 531344, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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 Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(398, 531467, F3, 12) (dual of [531467, 531369, 13]-code), using
- OOA 2-folding [i] based on linear OA(399, 531468, F3, 12) (dual of [531468, 531369, 13]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(399, 265734, F3, 2, 12) (dual of [(265734, 2), 531369, 13]-NRT-code), using
(89, 89+12, large)-Net in Base 3 — Upper bound on s
There is no (89, 101, large)-net in base 3, because
- 10 times m-reduction [i] would yield (89, 91, large)-net in base 3, but