Best Known (75, 75+12, s)-Nets in Base 16
(75, 75+12, 2797583)-Net over F16 — Constructive and digital
Digital (75, 87, 2797583)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 19, 1383)-net over F16, using
- (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 (10, 16, 1366)-net over F16, using
- net defined by OOA [i] based on linear OOA(1616, 1366, F16, 6, 6) (dual of [(1366, 6), 8180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1616, 4098, F16, 6) (dual of [4098, 4082, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1616, 4096, F16, 6) (dual of [4096, 4080, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1613, 4096, F16, 5) (dual of [4096, 4083, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1616, 4098, F16, 6) (dual of [4098, 4082, 7]-code), using
- net defined by OOA [i] based on linear OOA(1616, 1366, F16, 6, 6) (dual of [(1366, 6), 8180, 7]-NRT-code), using
- digital (0, 3, 17)-net over F16, using
- (u, u+v)-construction [i] based on
- 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 (13, 19, 1383)-net over F16, using
(75, 75+12, large)-Net over F16 — Digital
Digital (75, 87, large)-net over F16, using
- 5 times m-reduction [i] based on digital (75, 92, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1692, large, F16, 17) (dual of [large, large−92, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 1 times code embedding in larger space [i] based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1692, large, F16, 17) (dual of [large, large−92, 18]-code), using
(75, 75+12, large)-Net in Base 16 — Upper bound on s
There is no (75, 87, large)-net in base 16, because
- 10 times m-reduction [i] would yield (75, 77, large)-net in base 16, but