Best Known (82−11, 82, s)-Nets in Base 3
(82−11, 82, 35434)-Net over F3 — Constructive and digital
Digital (71, 82, 35434)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 35434, F3, 11, 11) (dual of [(35434, 11), 389692, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(382, 177171, F3, 11) (dual of [177171, 177089, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 177173, F3, 11) (dual of [177173, 177091, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 177173, F3, 11) (dual of [177173, 177091, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(382, 177171, F3, 11) (dual of [177171, 177089, 12]-code), using
(82−11, 82, 88586)-Net over F3 — Digital
Digital (71, 82, 88586)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(382, 88586, F3, 2, 11) (dual of [(88586, 2), 177090, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(382, 177172, F3, 11) (dual of [177172, 177090, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 177173, F3, 11) (dual of [177173, 177091, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 177173, F3, 11) (dual of [177173, 177091, 12]-code), using
- OOA 2-folding [i] based on linear OA(382, 177172, F3, 11) (dual of [177172, 177090, 12]-code), using
(82−11, 82, large)-Net in Base 3 — Upper bound on s
There is no (71, 82, large)-net in base 3, because
- 9 times m-reduction [i] would yield (71, 73, large)-net in base 3, but