Best Known (69, 69+11, s)-Nets in Base 32
(69, 69+11, 5033160)-Net over F32 — Constructive and digital
Digital (69, 80, 5033160)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (5, 8, 1677720)-net over F32, using
- s-reduction based on digital (5, 8, 3214400)-net over F32, using
- net defined by OOA [i] based on linear OOA(328, 3214400, F32, 3, 3) (dual of [(3214400, 3), 9643192, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(328, 3214400, F32, 2, 3) (dual of [(3214400, 2), 6428792, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(328, 3214400, F32, 3, 3) (dual of [(3214400, 3), 9643192, 4]-NRT-code), using
- s-reduction based on digital (5, 8, 3214400)-net over F32, using
- digital (16, 21, 1677720)-net over F32, using
- s-reduction 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
- s-reduction based on digital (16, 21, 4194301)-net over F32, using
- digital (40, 51, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (5, 8, 1677720)-net over F32, using
(69, 69+11, large)-Net over F32 — Digital
Digital (69, 80, large)-net over F32, using
- t-expansion [i] based on digital (68, 80, large)-net over F32, using
- 6 times m-reduction [i] based on digital (68, 86, large)-net over F32, using
(69, 69+11, large)-Net in Base 32 — Upper bound on s
There is no (69, 80, large)-net in base 32, because
- 9 times m-reduction [i] would yield (69, 71, large)-net in base 32, but