MinT - Best Known (58, 78, s)-Nets in Base 8

Best Known (58, 78, s)-Nets in Base 8

Digital (58, 78, 819)-net over F₈, using

net defined by OOA [i] based on linear OOA(8⁷⁸, 819, F₈, 20, 20) (dual of [(819, 20), 16302, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8⁷⁸, 8190, F₈, 20) (dual of [8190, 8112, 21]-code), using
  - discarding factors / shortening the dual code based on linear OA(8⁷⁸, 8192, F₈, 20) (dual of [8192, 8114, 21]-code), using
    - trace code [i] based on linear OA(64³⁹, 4096, F₆₄, 20) (dual of [4096, 4057, 21]-code), using
      - an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

Digital (58, 78, 7864)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁸, 7864, F₈, 20) (dual of [7864, 7786, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁸, 8192, F₈, 20) (dual of [8192, 8114, 21]-code), using
  - trace code [i] based on linear OA(64³⁹, 4096, F₆₄, 20) (dual of [4096, 4057, 21]-code), using
    - an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

There is no (58, 78, 7161102)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27607 021894 154289 870350 478333 520029 474477 145299 849330 022424 430871 190734 > 8⁷⁸ [i]