Best Known (68−11, 68, s)-Nets in Base 16
(68−11, 68, 3355464)-Net over F16 — Constructive and digital
Digital (57, 68, 3355464)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, 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 (1, 6, 24)-net over F16, using
(68−11, 68, large)-Net over F16 — Digital
Digital (57, 68, large)-net over F16, using
- 161 times duplication [i] based on digital (56, 67, large)-net over F16, using
- t-expansion [i] based on digital (55, 67, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1667, large, F16, 12) (dual of [large, large−67, 13]-code), using
- t-expansion [i] based on digital (55, 67, large)-net over F16, using
(68−11, 68, large)-Net in Base 16 — Upper bound on s
There is no (57, 68, large)-net in base 16, because
- 9 times m-reduction [i] would yield (57, 59, large)-net in base 16, but