Best Known (110−16, 110, s)-Nets in Base 3
(110−16, 110, 7388)-Net over F3 — Constructive and digital
Digital (94, 110, 7388)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 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 (85, 101, 7381)-net over F3, using
- net defined by OOA [i] based on linear OOA(3101, 7381, F3, 16, 16) (dual of [(7381, 16), 117995, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3101, 59048, F3, 16) (dual of [59048, 58947, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3101, 59048, F3, 16) (dual of [59048, 58947, 17]-code), using
- net defined by OOA [i] based on linear OOA(3101, 7381, F3, 16, 16) (dual of [(7381, 16), 117995, 17]-NRT-code), using
- digital (1, 9, 7)-net over F3, using
(110−16, 110, 26059)-Net over F3 — Digital
Digital (94, 110, 26059)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3110, 26059, F3, 2, 16) (dual of [(26059, 2), 52008, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3110, 29544, F3, 2, 16) (dual of [(29544, 2), 58978, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3110, 59088, F3, 16) (dual of [59088, 58978, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(38, 38, F3, 4) (dual of [38, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3109, 59087, F3, 16) (dual of [59087, 58978, 17]-code), using
- OOA 2-folding [i] based on linear OA(3110, 59088, F3, 16) (dual of [59088, 58978, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(3110, 29544, F3, 2, 16) (dual of [(29544, 2), 58978, 17]-NRT-code), using
(110−16, 110, 6840326)-Net in Base 3 — Upper bound on s
There is no (94, 110, 6840327)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30432 537067 613089 891553 398838 555870 606067 516194 072433 > 3110 [i]