Best Known (69, 80, s)-Nets in Base 3
(69, 80, 35432)-Net over F3 — Constructive and digital
Digital (69, 80, 35432)-net over F3, using
- net defined by OOA [i] based on linear OOA(380, 35432, F3, 11, 11) (dual of [(35432, 11), 389672, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(380, 177161, F3, 11) (dual of [177161, 177081, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(379, 177160, F3, 11) (dual of [177160, 177081, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(10) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(379, 177160, F3, 11) (dual of [177160, 177081, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(380, 177161, F3, 11) (dual of [177161, 177081, 12]-code), using
(69, 80, 84442)-Net over F3 — Digital
Digital (69, 80, 84442)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(380, 84442, F3, 2, 11) (dual of [(84442, 2), 168804, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(380, 88580, F3, 2, 11) (dual of [(88580, 2), 177080, 12]-NRT-code), using
- 31 times duplication [i] based on linear OOA(379, 88580, F3, 2, 11) (dual of [(88580, 2), 177081, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(379, 177160, F3, 11) (dual of [177160, 177081, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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(10) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(379, 177160, F3, 11) (dual of [177160, 177081, 12]-code), using
- 31 times duplication [i] based on linear OOA(379, 88580, F3, 2, 11) (dual of [(88580, 2), 177081, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(380, 88580, F3, 2, 11) (dual of [(88580, 2), 177080, 12]-NRT-code), using
(69, 80, large)-Net in Base 3 — Upper bound on s
There is no (69, 80, large)-net in base 3, because
- 9 times m-reduction [i] would yield (69, 71, large)-net in base 3, but