MinT - Best Known (85, 85+20, s)-Nets in Base 4

Best Known (85, 85+20, s)-Nets in Base 4

Digital (85, 105, 1638)-net over F₄, using

net defined by OOA [i] based on linear OOA(4¹⁰⁵, 1638, F₄, 20, 20) (dual of [(1638, 20), 32655, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4¹⁰⁵, 16380, F₄, 20) (dual of [16380, 16275, 21]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹⁰⁵, 16383, F₄, 20) (dual of [16383, 16278, 21]-code), using
    - 1 times truncation [i] based on linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

Digital (85, 105, 8191)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁰⁵, 8191, F₄, 2, 20) (dual of [(8191, 2), 16277, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4¹⁰⁵, 16382, F₄, 20) (dual of [16382, 16277, 21]-code), using
  - discarding factors / shortening the dual code based on linear OA(4¹⁰⁵, 16383, F₄, 20) (dual of [16383, 16278, 21]-code), using
    - 1 times truncation [i] based on linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using
      - an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

There is no (85, 105, 3165803)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 504685 940356 566771 557947 856311 473265 792936 248968 776001 317341 > 4¹⁰⁵ [i]