Best Known (14, 14+9, s)-Nets in Base 5
(14, 14+9, 64)-Net over F5 — Constructive and digital
Digital (14, 23, 64)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (9, 18, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 9, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 9, 26)-net over F25, using
(14, 14+9, 129)-Net over F5 — Digital
Digital (14, 23, 129)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(523, 129, F5, 9) (dual of [129, 106, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(523, 132, F5, 9) (dual of [132, 109, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(522, 125, F5, 9) (dual of [125, 103, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(516, 125, F5, 7) (dual of [125, 109, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(523, 132, F5, 9) (dual of [132, 109, 10]-code), using
(14, 14+9, 3864)-Net in Base 5 — Upper bound on s
There is no (14, 23, 3865)-net in base 5, because
- 1 times m-reduction [i] would yield (14, 22, 3865)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 2386 435620 537201 > 522 [i]