Best Known (92−29, 92, s)-Nets in Base 5
(92−29, 92, 258)-Net over F5 — Constructive and digital
Digital (63, 92, 258)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 14, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (49, 78, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
- digital (0, 14, 6)-net over F5, using
(92−29, 92, 602)-Net over F5 — Digital
Digital (63, 92, 602)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(592, 602, F5, 29) (dual of [602, 510, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(592, 634, F5, 29) (dual of [634, 542, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(591, 625, F5, 29) (dual of [625, 534, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(583, 625, F5, 27) (dual of [625, 542, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 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
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(592, 634, F5, 29) (dual of [634, 542, 30]-code), using
(92−29, 92, 52798)-Net in Base 5 — Upper bound on s
There is no (63, 92, 52799)-net in base 5, because
- 1 times m-reduction [i] would yield (63, 91, 52799)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4039 639950 268898 887346 958139 640766 036003 571551 215026 494026 598345 > 591 [i]