MinT - Best Known (90−12, 90, s)-Nets in Base 4

Best Known (90−12, 90, s)-Nets in Base 4

Digital (78, 90, 174762)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁹⁰, 174762, F₄, 12, 12) (dual of [(174762, 12), 2097054, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4⁹⁰, 1048572, F₄, 12) (dual of [1048572, 1048482, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁹⁰, 1048575, F₄, 12) (dual of [1048575, 1048485, 13]-code), using
    - 1 times truncation [i] based on linear OA(4⁹¹, 1048576, F₄, 13) (dual of [1048576, 1048485, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

Digital (78, 90, 524287)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4⁹⁰, 524287, F₄, 2, 12) (dual of [(524287, 2), 1048484, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4⁹⁰, 1048574, F₄, 12) (dual of [1048574, 1048484, 13]-code), using
  - discarding factors / shortening the dual code based on linear OA(4⁹⁰, 1048575, F₄, 12) (dual of [1048575, 1048485, 13]-code), using
    - 1 times truncation [i] based on linear OA(4⁹¹, 1048576, F₄, 13) (dual of [1048576, 1048485, 14]-code), using
      - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 4¹⁰âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

There is no (78, 90, large)-net in base 4, because

10 times m-reduction [i] would yield (78, 80, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]