Best Known (50, 50+14, s)-Nets in Base 16
(50, 50+14, 18763)-Net over F16 — Constructive and digital
Digital (50, 64, 18763)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-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 (3, 10, 38)-net over F16, using
(50, 50+14, 37450)-Net in Base 16 — Constructive
(50, 64, 37450)-net in base 16, using
- 161 times duplication [i] based on (49, 63, 37450)-net in base 16, using
- base change [i] based on digital (28, 42, 37450)-net over F64, using
- 641 times duplication [i] based on digital (27, 41, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 37450, F64, 14, 14) (dual of [(37450, 14), 524259, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6441, 262150, F64, 14) (dual of [262150, 262109, 15]-code), using
- discarding factors / shortening the dual code 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 factors / shortening the dual code based on linear OA(6441, 262151, F64, 14) (dual of [262151, 262110, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6441, 262150, F64, 14) (dual of [262150, 262109, 15]-code), using
- net defined by OOA [i] based on linear OOA(6441, 37450, F64, 14, 14) (dual of [(37450, 14), 524259, 15]-NRT-code), using
- 641 times duplication [i] based on digital (27, 41, 37450)-net over F64, using
- base change [i] based on digital (28, 42, 37450)-net over F64, using
(50, 50+14, 320110)-Net over F16 — Digital
Digital (50, 64, 320110)-net over F16, using
(50, 50+14, large)-Net in Base 16 — Upper bound on s
There is no (50, 64, large)-net in base 16, because
- 12 times m-reduction [i] would yield (50, 52, large)-net in base 16, but