Best Known (66, 66+10, s)-Nets in Base 3
(66, 66+10, 106291)-Net over F3 — Constructive and digital
Digital (66, 76, 106291)-net over F3, using
- 32 times duplication [i] based on digital (64, 74, 106291)-net over F3, using
- net defined by OOA [i] based on linear OOA(374, 106291, F3, 10, 10) (dual of [(106291, 10), 1062836, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- net defined by OOA [i] based on linear OOA(374, 106291, F3, 10, 10) (dual of [(106291, 10), 1062836, 11]-NRT-code), using
(66, 66+10, 186951)-Net over F3 — Digital
Digital (66, 76, 186951)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(376, 186951, F3, 2, 10) (dual of [(186951, 2), 373826, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(376, 265728, F3, 2, 10) (dual of [(265728, 2), 531380, 11]-NRT-code), using
- 31 times duplication [i] based on linear OOA(375, 265728, F3, 2, 10) (dual of [(265728, 2), 531381, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(375, 531456, F3, 10) (dual of [531456, 531381, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- OOA 2-folding [i] based on linear OA(375, 531456, F3, 10) (dual of [531456, 531381, 11]-code), using
- 31 times duplication [i] based on linear OOA(375, 265728, F3, 2, 10) (dual of [(265728, 2), 531381, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(376, 265728, F3, 2, 10) (dual of [(265728, 2), 531380, 11]-NRT-code), using
(66, 66+10, large)-Net in Base 3 — Upper bound on s
There is no (66, 76, large)-net in base 3, because
- 8 times m-reduction [i] would yield (66, 68, large)-net in base 3, but