Best Known (109, 109+11, s)-Nets in Base 16
(109, 109+11, large)-Net over F16 — Constructive and digital
Digital (109, 120, large)-net over F16, using
- 1 times m-reduction [i] based on digital (109, 121, large)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (15, 19, 2796200)-net over F16, using
- s-reduction based on digital (15, 19, 4194301)-net over F16, using
- net defined by OOA [i] based on linear OOA(1619, 4194301, F16, 4, 4) (dual of [(4194301, 4), 16777185, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1619, 8388602, F16, 4) (dual of [8388602, 8388583, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(1619, large, F16, 4) (dual of [large, large−19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(1619, large, F16, 4) (dual of [large, large−19, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1619, 8388602, F16, 4) (dual of [8388602, 8388583, 5]-code), using
- net defined by OOA [i] based on linear OOA(1619, 4194301, F16, 4, 4) (dual of [(4194301, 4), 16777185, 5]-NRT-code), using
- s-reduction based on digital (15, 19, 4194301)-net over F16, using
- digital (25, 31, 2796200)-net over F16, using
- s-reduction based on digital (25, 31, 2796201)-net over F16, using
- net defined by OOA [i] based on linear OOA(1631, 2796201, F16, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1631, large, F16, 6) (dual of [large, large−31, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(1631, large, F16, 6) (dual of [large, large−31, 7]-code), using
- net defined by OOA [i] based on linear OOA(1631, 2796201, F16, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
- s-reduction based on digital (25, 31, 2796201)-net over F16, using
- digital (56, 68, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(25634, large, F256, 12) (dual of [large, large−34, 13]-code), using
- OOA 2-folding [i] based on linear OA(25634, 8388602, F256, 12) (dual of [8388602, 8388568, 13]-code), using
- trace code [i] based on linear OOA(25634, 4194301, F256, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1668, 8388602, F16, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1668, 8388601, F16, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1668, 2796200, F16, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
- digital (0, 3, 17)-net over F16, using
- generalized (u, u+v)-construction [i] based on
(109, 109+11, large)-Net in Base 16 — Upper bound on s
There is no (109, 120, large)-net in base 16, because
- 9 times m-reduction [i] would yield (109, 111, large)-net in base 16, but