Best Known (85, 85+17, s)-Nets in Base 5
(85, 85+17, 9778)-Net over F5 — Constructive and digital
Digital (85, 102, 9778)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 12)-net over F5, using
- net from sequence [i] based on digital (2, 11)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 2 and N(F) ≥ 12, using
- net from sequence [i] based on digital (2, 11)-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 (2, 10, 12)-net over F5, using
(85, 85+17, 78171)-Net over F5 — Digital
Digital (85, 102, 78171)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5102, 78171, F5, 17) (dual of [78171, 78069, 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, 78170, F5, 16) (dual of [78170, 78069, 17]-code), using Gilbert–Varšamov bound and bm = 5101 > Vbs−1(k−1) = 20387 127563 782927 812732 496289 233456 957628 837709 569724 871989 290247 605029 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
(85, 85+17, large)-Net in Base 5 — Upper bound on s
There is no (85, 102, large)-net in base 5, because
- 15 times m-reduction [i] would yield (85, 87, large)-net in base 5, but