Best Known (68, 68+11, s)-Nets in Base 32
(68, 68+11, 4406064)-Net over F32 — Constructive and digital
Digital (68, 79, 4406064)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 7, 1050624)-net over F32, using
- net defined by OOA [i] based on linear OOA(327, 1050624, F32, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(327, 1050624, F32, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(327, 1050624, F32, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), 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 (4, 7, 1050624)-net over F32, using
(68, 68+11, 5033160)-Net in Base 32 — Constructive
(68, 79, 5033160)-net in base 32, using
- 321 times duplication [i] based on (67, 78, 5033160)-net in base 32, using
- base change [i] based on digital (54, 65, 5033160)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 7, 1677720)-net over F64, using
- s-reduction based on digital (4, 7, large)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, large, F64, 3, 3), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(647, large, F64, 3) (dual of [large, large−7, 4]-code), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- net defined by OOA [i] based on linear OOA(647, large, F64, 3, 3), using
- s-reduction based on digital (4, 7, large)-net over F64, using
- digital (12, 17, 1677720)-net over F64, using
- s-reduction based on digital (12, 17, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- s-reduction based on digital (12, 17, 4194301)-net over F64, using
- digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, 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(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (4, 7, 1677720)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- base change [i] based on digital (54, 65, 5033160)-net over F64, using
(68, 68+11, large)-Net over F32 — Digital
Digital (68, 79, large)-net over F32, using
- 7 times m-reduction [i] based on digital (68, 86, large)-net over F32, using
(68, 68+11, large)-Net in Base 32 — Upper bound on s
There is no (68, 79, large)-net in base 32, because
- 9 times m-reduction [i] would yield (68, 70, large)-net in base 32, but