Best Known (72, 72+28, s)-Nets in Base 16
(72, 72+28, 1093)-Net over F16 — Constructive and digital
Digital (72, 100, 1093)-net over F16, using
- 1 times m-reduction [i] based on digital (72, 101, 1093)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (6, 15, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- digital (29, 58, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 29, 257)-net over F256, using
- digital (6, 15, 65)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(72, 72+28, 1170)-Net in Base 16 — Constructive
(72, 100, 1170)-net in base 16, using
- t-expansion [i] based on (71, 100, 1170)-net in base 16, using
- net defined by OOA [i] based on OOA(16100, 1170, S16, 29, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(16100, 16381, S16, 29), using
- discarding factors based on OA(16100, 16386, S16, 29), using
- discarding parts of the base [i] based on linear OA(12857, 16386, F128, 29) (dual of [16386, 16329, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(12857, 16384, F128, 29) (dual of [16384, 16327, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(12855, 16384, F128, 28) (dual of [16384, 16329, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- discarding parts of the base [i] based on linear OA(12857, 16386, F128, 29) (dual of [16386, 16329, 30]-code), using
- discarding factors based on OA(16100, 16386, S16, 29), using
- OOA 14-folding and stacking with additional row [i] based on OA(16100, 16381, S16, 29), using
- net defined by OOA [i] based on OOA(16100, 1170, S16, 29, 29), using
(72, 72+28, 21004)-Net over F16 — Digital
Digital (72, 100, 21004)-net over F16, using
(72, 72+28, large)-Net in Base 16 — Upper bound on s
There is no (72, 100, large)-net in base 16, because
- 26 times m-reduction [i] would yield (72, 74, large)-net in base 16, but