Best Known (34, 44, s)-Nets in Base 16
(34, 44, 26239)-Net over F16 — Constructive and digital
Digital (34, 44, 26239)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 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 (28, 38, 26215)-net over F16, using
- net defined by OOA [i] based on linear OOA(1638, 26215, F16, 10, 10) (dual of [(26215, 10), 262112, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1638, 131075, F16, 10) (dual of [131075, 131037, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1638, 131075, F16, 10) (dual of [131075, 131037, 11]-code), using
- net defined by OOA [i] based on linear OOA(1638, 26215, F16, 10, 10) (dual of [(26215, 10), 262112, 11]-NRT-code), using
- digital (1, 6, 24)-net over F16, using
(34, 44, 52430)-Net in Base 16 — Constructive
(34, 44, 52430)-net in base 16, using
- net defined by OOA [i] based on OOA(1644, 52430, S16, 10, 10), using
- OA 5-folding and stacking [i] based on OA(1644, 262150, S16, 10), using
- discarding factors based on OA(1644, 262151, S16, 10), using
- discarding parts of the base [i] based on linear OA(6429, 262151, F64, 10) (dual of [262151, 262122, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 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(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(9) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(6429, 262151, F64, 10) (dual of [262151, 262122, 11]-code), using
- discarding factors based on OA(1644, 262151, S16, 10), using
- OA 5-folding and stacking [i] based on OA(1644, 262150, S16, 10), using
(34, 44, 213048)-Net over F16 — Digital
Digital (34, 44, 213048)-net over F16, using
(34, 44, large)-Net in Base 16 — Upper bound on s
There is no (34, 44, large)-net in base 16, because
- 8 times m-reduction [i] would yield (34, 36, large)-net in base 16, but