MinT - Best Known (121−13, 121, s)-Nets in Base 2

Best Known (121−13, 121, s)-Nets in Base 2

Digital (108, 121, 174762)-net over F₂, using

net defined by OOA [i] based on linear OOA(2¹²¹, 174762, F₂, 13, 13) (dual of [(174762, 13), 2271785, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2¹²¹, 1048573, F₂, 13) (dual of [1048573, 1048452, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹²¹, 1048576, F₂, 13) (dual of [1048576, 1048455, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 2²⁰âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (108, 121, 209715)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹²¹, 209715, F₂, 5, 13) (dual of [(209715, 5), 1048454, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2¹²¹, 1048575, F₂, 13) (dual of [1048575, 1048454, 14]-code), using
  - discarding factors / shortening the dual code based on linear OA(2¹²¹, 1048576, F₂, 13) (dual of [1048576, 1048455, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 2²⁰âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (108, 121, 3139214)-net in base 2, because

1 times m-reduction [i] would yield (108, 120, 3139214)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 329229 877081 755613 664954 023867 287100 > 2¹²⁰ [i]