Best Known (86, 86+28, s)-Nets in Base 5
(86, 86+28, 400)-Net over F5 — Constructive and digital
Digital (86, 114, 400)-net over F5, using
- 8 times m-reduction [i] based on digital (86, 122, 400)-net over F5, using
- trace code for nets [i] based on digital (25, 61, 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, 61, 200)-net over F25, using
(86, 86+28, 2860)-Net over F5 — Digital
Digital (86, 114, 2860)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5114, 2860, F5, 28) (dual of [2860, 2746, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5114, 3139, F5, 28) (dual of [3139, 3025, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(23) [i] based on
- linear OA(5111, 3125, F5, 28) (dual of [3125, 3014, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(5101, 3125, F5, 26) (dual of [3125, 3024, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(51, 12, F5, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(27) ⊂ Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(5114, 3139, F5, 28) (dual of [3139, 3025, 29]-code), using
(86, 86+28, 743027)-Net in Base 5 — Upper bound on s
There is no (86, 114, 743028)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 48 148844 889236 064105 372635 411700 220084 606612 219923 080069 927457 699970 872660 799425 > 5114 [i]