Best Known (11−5, 11, s)-Nets in Base 16
(11−5, 11, 514)-Net over F16 — Constructive and digital
Digital (6, 11, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (6, 12, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 6, 257)-net over F256, using
(11−5, 11, 723)-Net over F16 — Digital
Digital (6, 11, 723)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1611, 723, F16, 5) (dual of [723, 712, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(161, 241, F16, 1) (dual of [241, 240, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(163, 241, F16, 2) (dual of [241, 238, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(163, 255, F16, 2) (dual of [255, 252, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(163, 255, F16, 2) (dual of [255, 252, 3]-code), using
- linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- linear OA(161, 241, F16, 1) (dual of [241, 240, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
(11−5, 11, 2016)-Net in Base 16 — Constructive
(6, 11, 2016)-net in base 16, using
- net defined by OOA [i] based on OOA(1611, 2016, S16, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(1611, 4033, S16, 5), using
- discarding parts of the base [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(1611, 4033, S16, 5), using
(11−5, 11, 98860)-Net in Base 16 — Upper bound on s
There is no (6, 11, 98861)-net in base 16, because
- 1 times m-reduction [i] would yield (6, 10, 98861)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 1 099532 536306 > 1610 [i]