Best Known (87, 87+13, s)-Nets in Base 3
(87, 87+13, 88576)-Net over F3 — Constructive and digital
Digital (87, 100, 88576)-net over F3, using
- net defined by OOA [i] based on linear OOA(3100, 88576, F3, 13, 13) (dual of [(88576, 13), 1151388, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3100, 531457, F3, 13) (dual of [531457, 531357, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(398, 531455, F3, 13) (dual of [531455, 531357, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [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(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(10) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(398, 531455, F3, 13) (dual of [531455, 531357, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3100, 531457, F3, 13) (dual of [531457, 531357, 14]-code), using
(87, 87+13, 177152)-Net over F3 — Digital
Digital (87, 100, 177152)-net over F3, using
- 31 times duplication [i] based on digital (86, 99, 177152)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(399, 177152, F3, 3, 13) (dual of [(177152, 3), 531357, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(399, 531456, F3, 13) (dual of [531456, 531357, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(398, 531455, F3, 13) (dual of [531455, 531357, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [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(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(398, 531455, F3, 13) (dual of [531455, 531357, 14]-code), using
- OOA 3-folding [i] based on linear OA(399, 531456, F3, 13) (dual of [531456, 531357, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(399, 177152, F3, 3, 13) (dual of [(177152, 3), 531357, 14]-NRT-code), using
(87, 87+13, large)-Net in Base 3 — Upper bound on s
There is no (87, 100, large)-net in base 3, because
- 11 times m-reduction [i] would yield (87, 89, large)-net in base 3, but