Best Known (87, 87+15, s)-Nets in Base 16
(87, 87+15, 2397258)-Net over F16 — Constructive and digital
Digital (87, 102, 2397258)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 8, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 8, 258)-net over F256, using
- digital (71, 86, 2396742)-net over F16, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (9, 16, 516)-net over F16, using
(87, 87+15, large)-Net over F16 — Digital
Digital (87, 102, large)-net over F16, using
- t-expansion [i] based on digital (85, 102, large)-net over F16, using
- 2 times m-reduction [i] based on digital (85, 104, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16104, large, F16, 19) (dual of [large, large−104, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 1 times code embedding in larger space [i] based on linear OA(16103, large, F16, 19) (dual of [large, large−103, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16104, large, F16, 19) (dual of [large, large−104, 20]-code), using
- 2 times m-reduction [i] based on digital (85, 104, large)-net over F16, using
(87, 87+15, large)-Net in Base 16 — Upper bound on s
There is no (87, 102, large)-net in base 16, because
- 13 times m-reduction [i] would yield (87, 89, large)-net in base 16, but