Best Known (87−11, 87, s)-Nets in Base 16
(87−11, 87, 6710880)-Net over F16 — Constructive and digital
Digital (76, 87, 6710880)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (20, 25, 4194301)-net over F16, using
- net defined by OOA [i] based on linear OOA(1625, 4194301, F16, 5, 5) (dual of [(4194301, 5), 20971480, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(1625, large, F16, 5) (dual of [large, large−25, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(1625, large, F16, 5) (dual of [large, large−25, 6]-code), using
- net defined by OOA [i] based on linear OOA(1625, 4194301, F16, 5, 5) (dual of [(4194301, 5), 20971480, 6]-NRT-code), using
- digital (51, 62, 3355440)-net over F16, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- digital (20, 25, 4194301)-net over F16, using
(87−11, 87, large)-Net over F16 — Digital
Digital (76, 87, large)-net over F16, using
- t-expansion [i] based on 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
- 5 times m-reduction [i] based on digital (75, 92, large)-net over F16, using
(87−11, 87, large)-Net in Base 16 — Upper bound on s
There is no (76, 87, large)-net in base 16, because
- 9 times m-reduction [i] would yield (76, 78, large)-net in base 16, but