Best Known (79−11, 79, s)-Nets in Base 3
(79−11, 79, 35431)-Net over F3 — Constructive and digital
Digital (68, 79, 35431)-net over F3, using
- 31 times duplication [i] based on digital (67, 78, 35431)-net over F3, using
- net defined by OOA [i] based on linear OOA(378, 35431, F3, 11, 11) (dual of [(35431, 11), 389663, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(378, 177156, F3, 11) (dual of [177156, 177078, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(378, 177158, F3, 11) (dual of [177158, 177080, 12]-code), using
- construction X 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(30, 11, F3, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(378, 177158, F3, 11) (dual of [177158, 177080, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(378, 177156, F3, 11) (dual of [177156, 177078, 12]-code), using
- net defined by OOA [i] based on linear OOA(378, 35431, F3, 11, 11) (dual of [(35431, 11), 389663, 12]-NRT-code), using
(79−11, 79, 73606)-Net over F3 — Digital
Digital (68, 79, 73606)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(379, 73606, F3, 2, 11) (dual of [(73606, 2), 147133, 12]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OOA(379, 88580, F3, 2, 11) (dual of [(88580, 2), 177081, 12]-NRT-code), using
(79−11, 79, large)-Net in Base 3 — Upper bound on s
There is no (68, 79, large)-net in base 3, because
- 9 times m-reduction [i] would yield (68, 70, large)-net in base 3, but