Best Known (56, 56+17, s)-Nets in Base 32
(56, 56+17, 131105)-Net over F32 — Constructive and digital
Digital (56, 73, 131105)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 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 (48, 65, 131072)-net over F32, using
- net defined by OOA [i] based on linear OOA(3265, 131072, F32, 17, 17) (dual of [(131072, 17), 2228159, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 328−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(3265, 1048577, F32, 17) (dual of [1048577, 1048512, 18]-code), using
- net defined by OOA [i] based on linear OOA(3265, 131072, F32, 17, 17) (dual of [(131072, 17), 2228159, 18]-NRT-code), using
- digital (0, 8, 33)-net over F32, using
(56, 56+17, 262145)-Net in Base 32 — Constructive
(56, 73, 262145)-net in base 32, using
- 322 times duplication [i] based on (54, 71, 262145)-net in base 32, using
- net defined by OOA [i] based on OOA(3271, 262145, S32, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3271, 2097161, S32, 17), using
- 1 times code embedding in larger space [i] based on OA(3270, 2097160, S32, 17), using
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(12849, 2097153, F128, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(12843, 2097153, F128, 15) (dual of [2097153, 2097110, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding parts of the base [i] based on linear OA(12850, 2097160, F128, 17) (dual of [2097160, 2097110, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(3270, 2097160, S32, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3271, 2097161, S32, 17), using
- net defined by OOA [i] based on OOA(3271, 262145, S32, 17, 17), using
(56, 56+17, 1615943)-Net over F32 — Digital
Digital (56, 73, 1615943)-net over F32, using
(56, 56+17, large)-Net in Base 32 — Upper bound on s
There is no (56, 73, large)-net in base 32, because
- 15 times m-reduction [i] would yield (56, 58, large)-net in base 32, but