Best Known (45, 45+13, s)-Nets in Base 32
(45, 45+13, 174829)-Net over F32 — Constructive and digital
Digital (45, 58, 174829)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 66)-net over F32, using
- (u, u+v)-construction [i] based on
- 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, 6, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (36, 49, 174763)-net over F32, using
- net defined by OOA [i] based on linear OOA(3249, 174763, F32, 13, 13) (dual of [(174763, 13), 2271870, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3249, 1048579, F32, 13) (dual of [1048579, 1048530, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 1048580, F32, 13) (dual of [1048580, 1048531, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(3249, 1048576, F32, 13) (dual of [1048576, 1048527, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(3245, 1048576, F32, 12) (dual of [1048576, 1048531, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 1048580, F32, 13) (dual of [1048580, 1048531, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3249, 1048579, F32, 13) (dual of [1048579, 1048530, 14]-code), using
- net defined by OOA [i] based on linear OOA(3249, 174763, F32, 13, 13) (dual of [(174763, 13), 2271870, 14]-NRT-code), using
- digital (3, 9, 66)-net over F32, using
(45, 45+13, 349558)-Net in Base 32 — Constructive
(45, 58, 349558)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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
- (39, 52, 349525)-net in base 32, using
- net defined by OOA [i] based on OOA(3252, 349525, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3252, 2097151, S32, 13), using
- discarding factors based on OA(3252, 2097155, S32, 13), using
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(12837, 2097152, F128, 13) (dual of [2097152, 2097115, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(12834, 2097152, F128, 12) (dual of [2097152, 2097118, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- discarding parts of the base [i] based on linear OA(12837, 2097155, F128, 13) (dual of [2097155, 2097118, 14]-code), using
- discarding factors based on OA(3252, 2097155, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3252, 2097151, S32, 13), using
- net defined by OOA [i] based on OOA(3252, 349525, S32, 13, 13), using
- digital (0, 6, 33)-net over F32, using
(45, 45+13, 3212862)-Net over F32 — Digital
Digital (45, 58, 3212862)-net over F32, using
(45, 45+13, large)-Net in Base 32 — Upper bound on s
There is no (45, 58, large)-net in base 32, because
- 11 times m-reduction [i] would yield (45, 47, large)-net in base 32, but