Best Known (43−14, 43, s)-Nets in Base 5
(43−14, 43, 138)-Net over F5 — Constructive and digital
Digital (29, 43, 138)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (22, 36, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 18, 66)-net over F25, using
- digital (0, 7, 6)-net over F5, using
(43−14, 43, 315)-Net over F5 — Digital
Digital (29, 43, 315)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(543, 315, F5, 14) (dual of [315, 272, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(543, 313, F5, 14) (dual of [313, 270, 15]-code), using an extension Ce(13) of the narrow-sense BCH-code C(I) with length 312 | 54−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(541, 313, F5, 13) (dual of [313, 272, 14]-code), using an extension Ce(12) of the narrow-sense BCH-code C(I) with length 312 | 54−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
(43−14, 43, 16611)-Net in Base 5 — Upper bound on s
There is no (29, 43, 16612)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 137134 694060 485010 246074 427665 > 543 [i]