Best Known (67, 67+10, s)-Nets in Base 32
(67, 67+10, 6710944)-Net over F32 — Constructive and digital
Digital (67, 77, 6710944)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 209717)-net over F32, using
- s-reduction based on digital (0, 0, s)-net over F32 with arbitrarily large s, using
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 0, 209717)-net over F32 (see above)
- digital (0, 1, 209717)-net over F32, using
- s-reduction based on digital (0, 1, s)-net over F32 with arbitrarily large s, using
- digital (0, 1, 209717)-net over F32 (see above)
- digital (0, 1, 209717)-net over F32 (see above)
- digital (0, 1, 209717)-net over F32 (see above)
- digital (0, 1, 209717)-net over F32 (see above)
- digital (3, 5, 209717)-net over F32, using
- s-reduction based on digital (3, 5, 1082401)-net over F32, using
- digital (3, 5, 209717)-net over F32 (see above)
- digital (4, 7, 209717)-net over F32, using
- s-reduction 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
- s-reduction based on digital (4, 7, 1050624)-net over F32, using
- digital (12, 17, 209717)-net over F32, using
- s-reduction based on digital (12, 17, 524289)-net over F32, using
- net defined by OOA [i] based on linear OOA(3217, 524289, F32, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3217, 1048579, F32, 5) (dual of [1048579, 1048562, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(3217, 1048579, F32, 5) (dual of [1048579, 1048562, 6]-code), using
- net defined by OOA [i] based on linear OOA(3217, 524289, F32, 5, 5) (dual of [(524289, 5), 2621428, 6]-NRT-code), using
- s-reduction based on digital (12, 17, 524289)-net over F32, using
- digital (28, 38, 209717)-net over F32, using
- net defined by OOA [i] based on linear OOA(3238, 209717, F32, 10, 10) (dual of [(209717, 10), 2097132, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3238, 1048585, F32, 10) (dual of [1048585, 1048547, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3229, 1048576, F32, 8) (dual of [1048576, 1048547, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(3238, 1048585, F32, 10) (dual of [1048585, 1048547, 11]-code), using
- net defined by OOA [i] based on linear OOA(3238, 209717, F32, 10, 10) (dual of [(209717, 10), 2097132, 11]-NRT-code), using
- digital (0, 0, 209717)-net over F32, using
(67, 67+10, large)-Net over F32 — Digital
Digital (67, 77, large)-net over F32, using
- t-expansion [i] based on digital (64, 77, large)-net over F32, using
- 4 times m-reduction [i] based on digital (64, 81, large)-net over F32, using
(67, 67+10, large)-Net in Base 32 — Upper bound on s
There is no (67, 77, large)-net in base 32, because
- 8 times m-reduction [i] would yield (67, 69, large)-net in base 32, but