Best Known (63, 63+11, s)-Nets in Base 16
(63, 63+11, 3355971)-Net over F16 — Constructive and digital
Digital (63, 74, 3355971)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (7, 12, 531)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 17)-net over F16, using
- digital (5, 10, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 5, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 5, 257)-net over F256, using
- (u, u+v)-construction [i] based on
- 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 (7, 12, 531)-net over F16, using
(63, 63+11, 3357456)-Net in Base 16 — Constructive
(63, 74, 3357456)-net in base 16, using
- 161 times duplication [i] based on (62, 73, 3357456)-net in base 16, using
- (u, u+v)-construction [i] based on
- (6, 11, 2016)-net in base 16, using
- net defined by OOA [i] based on OOA(1611, 2016, S16, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(1611, 4033, S16, 5), using
- discarding parts of the base [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(1611, 4033, S16, 5), using
- net defined by OOA [i] based on OOA(1611, 2016, S16, 5, 5), 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
- (6, 11, 2016)-net in base 16, using
- (u, u+v)-construction [i] based on
(63, 63+11, large)-Net over F16 — Digital
Digital (63, 74, large)-net over F16, using
- 161 times duplication [i] based on digital (62, 73, large)-net over F16, using
- t-expansion [i] based on digital (60, 73, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1673, large, F16, 13) (dual of [large, large−73, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1673, large, F16, 13) (dual of [large, large−73, 14]-code), using
- t-expansion [i] based on digital (60, 73, large)-net over F16, using
(63, 63+11, large)-Net in Base 16 — Upper bound on s
There is no (63, 74, large)-net in base 16, because
- 9 times m-reduction [i] would yield (63, 65, large)-net in base 16, but