Best Known (55−15, 55, s)-Nets in Base 32
(55−15, 55, 4780)-Net over F32 — Constructive and digital
Digital (40, 55, 4780)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 99)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- digital (0, 3, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (0, 7, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- generalized (u, u+v)-construction [i] based on
- digital (28, 43, 4681)-net over F32, using
- net defined by OOA [i] based on linear OOA(3243, 4681, F32, 15, 15) (dual of [(4681, 15), 70172, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3243, 32768, F32, 15) (dual of [32768, 32725, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(3243, 32768, F32, 15) (dual of [32768, 32725, 16]-code), using
- net defined by OOA [i] based on linear OOA(3243, 4681, F32, 15, 15) (dual of [(4681, 15), 70172, 16]-NRT-code), using
- digital (5, 12, 99)-net over F32, using
(55−15, 55, 37450)-Net in Base 32 — Constructive
(40, 55, 37450)-net in base 32, using
- 321 times duplication [i] based on (39, 54, 37450)-net in base 32, using
- base change [i] based on digital (30, 45, 37450)-net over F64, using
- 641 times duplication [i] based on digital (29, 44, 37450)-net over F64, using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(6443, 262145, F64, 15) (dual of [262145, 262102, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(6437, 262145, F64, 13) (dual of [262145, 262108, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [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 C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6444, 262152, F64, 15) (dual of [262152, 262108, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6444, 262151, F64, 15) (dual of [262151, 262107, 16]-code), using
- net defined by OOA [i] based on linear OOA(6444, 37450, F64, 15, 15) (dual of [(37450, 15), 561706, 16]-NRT-code), using
- 641 times duplication [i] based on digital (29, 44, 37450)-net over F64, using
- base change [i] based on digital (30, 45, 37450)-net over F64, using
(55−15, 55, 159663)-Net over F32 — Digital
Digital (40, 55, 159663)-net over F32, using
(55−15, 55, large)-Net in Base 32 — Upper bound on s
There is no (40, 55, large)-net in base 32, because
- 13 times m-reduction [i] would yield (40, 42, large)-net in base 32, but