Best Known (109−27, 109, s)-Nets in Base 5
(109−27, 109, 400)-Net over F5 — Constructive and digital
Digital (82, 109, 400)-net over F5, using
- 5 times m-reduction [i] based on digital (82, 114, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 57, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- trace code for nets [i] based on digital (25, 57, 200)-net over F25, using
(109−27, 109, 2644)-Net over F5 — Digital
Digital (82, 109, 2644)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 2644, F5, 27) (dual of [2644, 2535, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(5109, 3138, F5, 27) (dual of [3138, 3029, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(5106, 3125, F5, 27) (dual of [3125, 3019, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(596, 3125, F5, 24) (dual of [3125, 3029, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(53, 13, F5, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5109, 3138, F5, 27) (dual of [3138, 3029, 28]-code), using
(109−27, 109, 908174)-Net in Base 5 — Upper bound on s
There is no (82, 109, 908175)-net in base 5, because
- 1 times m-reduction [i] would yield (82, 108, 908175)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3081 492593 540497 044685 913540 177554 997512 563420 210573 169911 044736 068763 406861 > 5108 [i]