Best Known (68, 68+22, s)-Nets in Base 5
(68, 68+22, 312)-Net over F5 — Constructive and digital
Digital (68, 90, 312)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (17, 28, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- digital (40, 62, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 31, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 31, 104)-net over F25, using
- digital (17, 28, 104)-net over F5, using
(68, 68+22, 2663)-Net over F5 — Digital
Digital (68, 90, 2663)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 2663, F5, 22) (dual of [2663, 2573, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(590, 3144, F5, 22) (dual of [3144, 3054, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(586, 3125, F5, 22) (dual of [3125, 3039, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(571, 3125, F5, 18) (dual of [3125, 3054, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(590, 3144, F5, 22) (dual of [3144, 3054, 23]-code), using
(68, 68+22, 642383)-Net in Base 5 — Upper bound on s
There is no (68, 90, 642384)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 807 799703 386079 017572 699905 076598 974675 554904 377604 300942 268225 > 590 [i]