Best Known (77, 77+14, s)-Nets in Base 16
(77, 77+14, 2396787)-Net over F16 — Constructive and digital
Digital (77, 91, 2396787)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 11, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- digital (4, 11, 45)-net over F16, using
(77, 77+14, 2396807)-Net in Base 16 — Constructive
(77, 91, 2396807)-net in base 16, using
- (u, u+v)-construction [i] based on
- (4, 11, 65)-net in base 16, using
- 1 times m-reduction [i] based on (4, 12, 65)-net in base 16, using
- base change [i] based on digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 8, 65)-net over F64, using
- 1 times m-reduction [i] based on (4, 12, 65)-net in base 16, using
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- (4, 11, 65)-net in base 16, using
(77, 77+14, large)-Net over F16 — Digital
Digital (77, 91, large)-net over F16, using
- t-expansion [i] based on digital (75, 91, large)-net over F16, using
- 1 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
- 1 times m-reduction [i] based on digital (75, 92, large)-net over F16, using
(77, 77+14, large)-Net in Base 16 — Upper bound on s
There is no (77, 91, large)-net in base 16, because
- 12 times m-reduction [i] would yield (77, 79, large)-net in base 16, but