Best Known (60, 60+11, s)-Nets in Base 3
(60, 60+11, 11811)-Net over F3 — Constructive and digital
Digital (60, 71, 11811)-net over F3, using
- net defined by OOA [i] based on linear OOA(371, 11811, F3, 11, 11) (dual of [(11811, 11), 129850, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(371, 59056, F3, 11) (dual of [59056, 58985, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(371, 59059, F3, 11) (dual of [59059, 58988, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(371, 59049, F3, 11) (dual of [59049, 58978, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(371, 59059, F3, 11) (dual of [59059, 58988, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(371, 59056, F3, 11) (dual of [59056, 58985, 12]-code), using
(60, 60+11, 24531)-Net over F3 — Digital
Digital (60, 71, 24531)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(371, 24531, F3, 2, 11) (dual of [(24531, 2), 48991, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(371, 29529, F3, 2, 11) (dual of [(29529, 2), 58987, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(371, 59058, F3, 11) (dual of [59058, 58987, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(371, 59059, F3, 11) (dual of [59059, 58988, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(371, 59049, F3, 11) (dual of [59049, 58978, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 10, F3, 0) (dual of [10, 10, 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(371, 59059, F3, 11) (dual of [59059, 58988, 12]-code), using
- OOA 2-folding [i] based on linear OA(371, 59058, F3, 11) (dual of [59058, 58987, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(371, 29529, F3, 2, 11) (dual of [(29529, 2), 58987, 12]-NRT-code), using
(60, 60+11, 6230221)-Net in Base 3 — Upper bound on s
There is no (60, 71, 6230222)-net in base 3, because
- 1 times m-reduction [i] would yield (60, 70, 6230222)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2503 155969 238899 337588 430820 494901 > 370 [i]