Best Known (90−17, 90, s)-Nets in Base 5
(90−17, 90, 1969)-Net over F5 — Constructive and digital
Digital (73, 90, 1969)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (62, 79, 1953)-net over F5, using
- net defined by OOA [i] based on linear OOA(579, 1953, F5, 17, 17) (dual of [(1953, 17), 33122, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using
- net defined by OOA [i] based on linear OOA(579, 1953, F5, 17, 17) (dual of [(1953, 17), 33122, 18]-NRT-code), using
- digital (3, 11, 16)-net over F5, using
(90−17, 90, 15668)-Net over F5 — Digital
Digital (73, 90, 15668)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 15668, F5, 17) (dual of [15668, 15578, 18]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(588, 15664, F5, 17) (dual of [15664, 15576, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(59, 39, F5, 5) (dual of [39, 30, 6]-code), using
- discarding factors / shortening the dual code based on 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(588, 15666, F5, 16) (dual of [15666, 15578, 17]-code), using Gilbert–Varšamov bound and bm = 588 > Vbs−1(k−1) = 684775 809621 949114 753214 447816 544653 717646 047568 928814 544965 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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(588, 15664, F5, 17) (dual of [15664, 15576, 18]-code), using
- construction X with Varšamov bound [i] based on
(90−17, 90, large)-Net in Base 5 — Upper bound on s
There is no (73, 90, large)-net in base 5, because
- 15 times m-reduction [i] would yield (73, 75, large)-net in base 5, but