Best Known (92, 92+31, s)-Nets in Base 5
(92, 92+31, 408)-Net over F5 — Constructive and digital
Digital (92, 123, 408)-net over F5, using
- 1 times m-reduction [i] based on digital (92, 124, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 62, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 62, 204)-net over F25, using
(92, 92+31, 2524)-Net over F5 — Digital
Digital (92, 123, 2524)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5123, 2524, F5, 31) (dual of [2524, 2401, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(5123, 3133, F5, 31) (dual of [3133, 3010, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(5121, 3125, F5, 31) (dual of [3125, 3004, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(5116, 3125, F5, 29) (dual of [3125, 3009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(51, 7, F5, 1) (dual of [7, 6, 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(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(5123, 3133, F5, 31) (dual of [3133, 3010, 32]-code), using
(92, 92+31, 777421)-Net in Base 5 — Upper bound on s
There is no (92, 123, 777422)-net in base 5, because
- 1 times m-reduction [i] would yield (92, 122, 777422)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 18 808005 095844 047480 765902 558251 984160 128907 666745 179212 948712 510507 639539 795144 040281 > 5122 [i]