Best Known (11−5, 11, s)-Nets in Base 8
(11−5, 11, 130)-Net over F8 — Constructive and digital
Digital (6, 11, 130)-net over F8, using
- 1 times m-reduction [i] based on digital (6, 12, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 6, 65)-net over F64, using
(11−5, 11, 171)-Net over F8 — Digital
Digital (6, 11, 171)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(811, 171, F8, 5) (dual of [171, 160, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(81, 57, F8, 1) (dual of [57, 56, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(83, 57, F8, 2) (dual of [57, 54, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- linear OA(81, 57, F8, 1) (dual of [57, 56, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
(11−5, 11, 6619)-Net in Base 8 — Upper bound on s
There is no (6, 11, 6620)-net in base 8, because
- 1 times m-reduction [i] would yield (6, 10, 6620)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1073 952671 > 810 [i]