Best Known (77−11, 77, s)-Nets in Base 3
(77−11, 77, 11818)-Net over F3 — Constructive and digital
Digital (66, 77, 11818)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- 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
- net defined by OOA [i] based on linear OOA(371, 11811, F3, 11, 11) (dual of [(11811, 11), 129850, 12]-NRT-code), using
- digital (1, 6, 7)-net over F3, using
(77−11, 77, 29542)-Net over F3 — Digital
Digital (66, 77, 29542)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(377, 29542, F3, 2, 11) (dual of [(29542, 2), 59007, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(377, 59084, F3, 11) (dual of [59084, 59007, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(377, 59085, F3, 11) (dual of [59085, 59008, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [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(341, 59049, F3, 7) (dual of [59049, 59008, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 36, F3, 3) (dual of [36, 30, 4]-code or 36-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(377, 59085, F3, 11) (dual of [59085, 59008, 12]-code), using
- OOA 2-folding [i] based on linear OA(377, 59084, F3, 11) (dual of [59084, 59007, 12]-code), using
(77−11, 77, large)-Net in Base 3 — Upper bound on s
There is no (66, 77, large)-net in base 3, because
- 9 times m-reduction [i] would yield (66, 68, large)-net in base 3, but