Best Known (104−17, 104, s)-Nets in Base 5
(104−17, 104, 9784)-Net over F5 — Constructive and digital
Digital (87, 104, 9784)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 18)-net over F5, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 4 and N(F) ≥ 18, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- digital (75, 92, 9766)-net over F5, using
- net defined by OOA [i] based on linear OOA(592, 9766, F5, 17, 17) (dual of [(9766, 17), 165930, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(592, 78129, F5, 17) (dual of [78129, 78037, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 78132, F5, 17) (dual of [78132, 78040, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 78132, F5, 17) (dual of [78132, 78040, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(592, 78129, F5, 17) (dual of [78129, 78037, 18]-code), using
- net defined by OOA [i] based on linear OOA(592, 9766, F5, 17, 17) (dual of [(9766, 17), 165930, 18]-NRT-code), using
- digital (4, 12, 18)-net over F5, using
(104−17, 104, 78175)-Net over F5 — Digital
Digital (87, 104, 78175)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5104, 78175, F5, 17) (dual of [78175, 78071, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5101, 78169, F5, 17) (dual of [78169, 78068, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(5101, 78172, F5, 16) (dual of [78172, 78071, 17]-code), using Gilbert–Varšamov bound and bm = 5101 > Vbs−1(k−1) = 20394 953891 574889 972015 583081 550216 877768 996882 849382 492448 088593 421981 [i]
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(5101, 78169, F5, 17) (dual of [78169, 78068, 18]-code), using
- construction X with Varšamov bound [i] based on
(104−17, 104, large)-Net in Base 5 — Upper bound on s
There is no (87, 104, large)-net in base 5, because
- 15 times m-reduction [i] would yield (87, 89, large)-net in base 5, but