Best Known (54, 54+15, s)-Nets in Base 32
(54, 54+15, 149896)-Net over F32 — Constructive and digital
Digital (54, 69, 149896)-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 (42, 57, 149797)-net over F32, using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- linear OA(3257, 1048576, F32, 15) (dual of [1048576, 1048519, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3253, 1048576, F32, 14) (dual of [1048576, 1048523, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(14) ⊂ Ce(13) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(3257, 1048580, F32, 15) (dual of [1048580, 1048523, 16]-code), using
- net defined by OOA [i] based on linear OOA(3257, 149797, F32, 15, 15) (dual of [(149797, 15), 2246898, 16]-NRT-code), using
- digital (5, 12, 99)-net over F32, using
(54, 54+15, 1198371)-Net in Base 32 — Constructive
(54, 69, 1198371)-net in base 32, using
- net defined by OOA [i] based on OOA(3269, 1198371, S32, 15, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
- discarding factors based on OA(3269, large, S32, 15), using
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- discarding factors based on OA(3269, large, S32, 15), using
- OOA 7-folding and stacking with additional row [i] based on OA(3269, 8388598, S32, 15), using
(54, 54+15, 5108999)-Net over F32 — Digital
Digital (54, 69, 5108999)-net over F32, using
(54, 54+15, large)-Net in Base 32 — Upper bound on s
There is no (54, 69, large)-net in base 32, because
- 13 times m-reduction [i] would yield (54, 56, large)-net in base 32, but