Best Known (57, 57+10, s)-Nets in Base 32
(57, 57+10, 3355440)-Net over F32 — Constructive and digital
Digital (57, 67, 3355440)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (16, 21, 4194301)-net over F32, using
- net defined by OOA [i] based on linear OOA(3221, 4194301, F32, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3221, large, F32, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(3221, large, F32, 5) (dual of [large, large−21, 6]-code), using
- net defined by OOA [i] based on linear OOA(3221, 4194301, F32, 5, 5) (dual of [(4194301, 5), 20971484, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (16, 21, 4194301)-net over F32, using
(57, 57+10, large)-Net over F32 — Digital
Digital (57, 67, large)-net over F32, using
- t-expansion [i] based on digital (56, 67, large)-net over F32, using
- 4 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- 4 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
(57, 57+10, large)-Net in Base 32 — Upper bound on s
There is no (57, 67, large)-net in base 32, because
- 8 times m-reduction [i] would yield (57, 59, large)-net in base 32, but