Best Known (48, 62, s)-Nets in Base 16
(48, 62, 18749)-Net over F16 — Constructive and digital
Digital (48, 62, 18749)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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 (40, 54, 18725)-net over F16, using
- net defined by OOA [i] based on linear OOA(1654, 18725, F16, 14, 14) (dual of [(18725, 14), 262096, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(25627, 65538, F256, 14) (dual of [65538, 65511, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(1654, 131076, F16, 14) (dual of [131076, 131022, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(1654, 131075, F16, 14) (dual of [131075, 131021, 15]-code), using
- net defined by OOA [i] based on linear OOA(1654, 18725, F16, 14, 14) (dual of [(18725, 14), 262096, 15]-NRT-code), using
- digital (1, 8, 24)-net over F16, using
(48, 62, 37450)-Net in Base 16 — Constructive
(48, 62, 37450)-net in base 16, using
- net defined by OOA [i] based on OOA(1662, 37450, S16, 14, 14), using
- OA 7-folding and stacking [i] based on OA(1662, 262150, S16, 14), using
- discarding factors based on OA(1662, 262151, S16, 14), using
- discarding parts of the base [i] based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(6440, 262144, F64, 14) (dual of [262144, 262104, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- discarding factors based on OA(1662, 262151, S16, 14), using
- OA 7-folding and stacking [i] based on OA(1662, 262150, S16, 14), using
(48, 62, 208956)-Net over F16 — Digital
Digital (48, 62, 208956)-net over F16, using
(48, 62, large)-Net in Base 16 — Upper bound on s
There is no (48, 62, large)-net in base 16, because
- 12 times m-reduction [i] would yield (48, 50, large)-net in base 16, but