Best Known (71−11, 71, s)-Nets in Base 16
(71−11, 71, 3355697)-Net over F16 — Constructive and digital
Digital (60, 71, 3355697)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 9, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) 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
- base reduction for projective spaces (embedding PG(4,256) in PG(8,16)) for nets [i] based on digital (0, 5, 257)-net over F256, 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 (4, 9, 257)-net over F16, using
(71−11, 71, 3355936)-Net in Base 16 — Constructive
(60, 71, 3355936)-net in base 16, using
- (u, u+v)-construction [i] based on
- (4, 9, 496)-net in base 16, using
- net defined by OOA [i] based on OOA(169, 496, S16, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(169, 993, S16, 5), using
- discarding parts of the base [i] based on linear OA(327, 993, F32, 5) (dual of [993, 986, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(169, 993, S16, 5), using
- net defined by OOA [i] based on OOA(169, 496, 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
- (4, 9, 496)-net in base 16, using
(71−11, 71, large)-Net over F16 — Digital
Digital (60, 71, large)-net over F16, using
- 2 times m-reduction [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
(71−11, 71, large)-Net in Base 16 — Upper bound on s
There is no (60, 71, large)-net in base 16, because
- 9 times m-reduction [i] would yield (60, 62, large)-net in base 16, but