MinT - Best Known (109−3, 109, s)-Nets in Base 4

Best Known (109−3, 109, s)-Nets in Base 4

(109−3, 109, large)-Net over F₄ — Constructive and digital

Digital (106, 109, large)-net over F₄, using

4⁵ times duplication [i] based on digital (101, 104, large)-net over F₄, using
- t-expansion [i] based on digital (96, 104, large)-net over F₄, using
  - trace code for nets [i] based on digital (18, 26, 2097406)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256²⁶, 2097406, F₂₅₆, 12, 8) (dual of [(2097406, 12), 25168846, 9]-NRT-code), using
      - OOA stacking with additional row [i] based on linear OOA(256²⁶, 2097407, F₂₅₆, 4, 8) (dual of [(2097407, 4), 8389602, 9]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(256⁴, 257, F₂₅₆, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;1024,256) [i]
        linear OOA(256²², 2097150, F₂₅₆, 4, 8) (dual of [(2097150, 4), 8388578, 9]-NRT-code), using
        OOA 4-folding [i] based on linear OA(256²², 8388600, F₂₅₆, 8) (dual of [8388600, 8388578, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(256²², large, F₂₅₆, 8) (dual of [large, large−22, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(109−3, 109, large)-Net in Base 4 — Upper bound on s

There is no (106, 109, large)-net in base 4, because

1 times m-reduction [i] would yield (106, 108, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]