Best Known (33, 33+10, s)-Nets in Base 16
(33, 33+10, 26232)-Net over F16 — Constructive and digital
Digital (33, 43, 26232)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-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 (0, 5, 17)-net over F16, using
(33, 33+10, 52429)-Net in Base 16 — Constructive
(33, 43, 52429)-net in base 16, using
- 161 times duplication [i] based on (32, 42, 52429)-net in base 16, using
- base change [i] based on digital (18, 28, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [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(6425, 262144, F64, 9) (dual of [262144, 262119, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(6428, 262147, F64, 10) (dual of [262147, 262119, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6428, 262145, F64, 10) (dual of [262145, 262117, 11]-code), using
- net defined by OOA [i] based on linear OOA(6428, 52429, F64, 10, 10) (dual of [(52429, 10), 524262, 11]-NRT-code), using
- base change [i] based on digital (18, 28, 52429)-net over F64, using
(33, 33+10, 156563)-Net over F16 — Digital
Digital (33, 43, 156563)-net over F16, using
(33, 33+10, large)-Net in Base 16 — Upper bound on s
There is no (33, 43, large)-net in base 16, because
- 8 times m-reduction [i] would yield (33, 35, large)-net in base 16, but