Best Known (27, 35, s)-Nets in Base 16
(27, 35, 32793)-Net over F16 — Constructive and digital
Digital (27, 35, 32793)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 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 (22, 30, 32769)-net over F16, using
- net defined by OOA [i] based on linear OOA(1630, 32769, F16, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1630, 131076, F16, 8) (dual of [131076, 131046, 9]-code), using
- trace code [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1630, 131076, F16, 8) (dual of [131076, 131046, 9]-code), using
- net defined by OOA [i] based on linear OOA(1630, 32769, F16, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
- digital (1, 5, 24)-net over F16, using
(27, 35, 65537)-Net in Base 16 — Constructive
(27, 35, 65537)-net in base 16, using
- 161 times duplication [i] based on (26, 34, 65537)-net in base 16, using
- net defined by OOA [i] based on OOA(1634, 65537, S16, 8, 8), using
- OA 4-folding and stacking [i] based on OA(1634, 262148, S16, 8), using
- 1 times code embedding in larger space [i] based on OA(1633, 262147, S16, 8), using
- discarding parts of the base [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- 1 times code embedding in larger space [i] based on OA(1633, 262147, S16, 8), using
- OA 4-folding and stacking [i] based on OA(1634, 262148, S16, 8), using
- net defined by OOA [i] based on OOA(1634, 65537, S16, 8, 8), using
(27, 35, 236284)-Net over F16 — Digital
Digital (27, 35, 236284)-net over F16, using
(27, 35, large)-Net in Base 16 — Upper bound on s
There is no (27, 35, large)-net in base 16, because
- 6 times m-reduction [i] would yield (27, 29, large)-net in base 16, but