Best Known (72−16, 72, s)-Nets in Base 16
(72−16, 72, 16417)-Net over F16 — Constructive and digital
Digital (56, 72, 16417)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (46, 62, 16384)-net over F16, using
- net defined by OOA [i] based on linear OOA(1662, 16384, F16, 16, 16) (dual of [(16384, 16), 262082, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(1662, 131072, F16, 16) (dual of [131072, 131010, 17]-code), using
- trace code [i] based on linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- trace code [i] based on linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(1662, 131072, F16, 16) (dual of [131072, 131010, 17]-code), using
- net defined by OOA [i] based on linear OOA(1662, 16384, F16, 16, 16) (dual of [(16384, 16), 262082, 17]-NRT-code), using
- digital (2, 10, 33)-net over F16, using
(72−16, 72, 32769)-Net in Base 16 — Constructive
(56, 72, 32769)-net in base 16, using
- base change [i] based on digital (32, 48, 32769)-net over F64, using
- net defined by OOA [i] based on linear OOA(6448, 32769, F64, 16, 16) (dual of [(32769, 16), 524256, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6448, 262152, F64, 16) (dual of [262152, 262104, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6448, 262155, F64, 16) (dual of [262155, 262107, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(6446, 262144, F64, 16) (dual of [262144, 262098, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(642, 11, F64, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(6448, 262155, F64, 16) (dual of [262155, 262107, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6448, 262152, F64, 16) (dual of [262152, 262104, 17]-code), using
- net defined by OOA [i] based on linear OOA(6448, 32769, F64, 16, 16) (dual of [(32769, 16), 524256, 17]-NRT-code), using
(72−16, 72, 257907)-Net over F16 — Digital
Digital (56, 72, 257907)-net over F16, using
(72−16, 72, large)-Net in Base 16 — Upper bound on s
There is no (56, 72, large)-net in base 16, because
- 14 times m-reduction [i] would yield (56, 58, large)-net in base 16, but