Best Known (44, 44+13, s)-Nets in Base 16
(44, 44+13, 21869)-Net over F16 — Constructive and digital
Digital (44, 57, 21869)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 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 (37, 50, 21845)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(1650, 131074, F16, 13) (dual of [131074, 131024, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(1650, 131071, F16, 13) (dual of [131071, 131021, 14]-code), using
- net defined by OOA [i] based on linear OOA(1650, 21845, F16, 13, 13) (dual of [(21845, 13), 283935, 14]-NRT-code), using
- digital (1, 7, 24)-net over F16, using
(44, 44+13, 43691)-Net in Base 16 — Constructive
(44, 57, 43691)-net in base 16, using
- base change [i] based on digital (25, 38, 43691)-net over F64, using
- 641 times duplication [i] based on digital (24, 37, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(6437, 262144, F64, 13) (dual of [262144, 262107, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(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(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(6437, 262147, F64, 13) (dual of [262147, 262110, 14]-code), using
- net defined by OOA [i] based on linear OOA(6437, 43691, F64, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
- 641 times duplication [i] based on digital (24, 37, 43691)-net over F64, using
(44, 44+13, 184865)-Net over F16 — Digital
Digital (44, 57, 184865)-net over F16, using
(44, 44+13, large)-Net in Base 16 — Upper bound on s
There is no (44, 57, large)-net in base 16, because
- 11 times m-reduction [i] would yield (44, 46, large)-net in base 16, but