Best Known (57, 74, s)-Nets in Base 32
(57, 74, 131116)-Net over F32 — Constructive and digital
Digital (57, 74, 131116)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-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 (1, 9, 44)-net over F32, using
(57, 74, 262146)-Net in Base 32 — Constructive
(57, 74, 262146)-net in base 32, using
- net defined by OOA [i] based on OOA(3274, 262146, S32, 17, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3274, 2097169, S32, 17), using
- 1 times code embedding in larger space [i] based on OA(3273, 2097168, S32, 17), using
- discarding parts of the base [i] based on linear OA(12852, 2097168, F128, 17) (dual of [2097168, 2097116, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [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(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding parts of the base [i] based on linear OA(12852, 2097168, F128, 17) (dual of [2097168, 2097116, 18]-code), using
- 1 times code embedding in larger space [i] based on OA(3273, 2097168, S32, 17), using
- OOA 8-folding and stacking with additional row [i] based on OA(3274, 2097169, S32, 17), using
(57, 74, 2006769)-Net over F32 — Digital
Digital (57, 74, 2006769)-net over F32, using
(57, 74, large)-Net in Base 32 — Upper bound on s
There is no (57, 74, large)-net in base 32, because
- 15 times m-reduction [i] would yield (57, 59, large)-net in base 32, but