MinT - Best Known (68, 68+8, s)-Nets in Base 4

Best Known (68, 68+8, s)-Nets in Base 4

(68, 68+8, 2097155)-Net over F₄ — Constructive and digital

Digital (68, 76, 2097155)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 4, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (64, 72, 2097150)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4⁷², 2097150, F₄, 8, 8) (dual of [(2097150, 8), 16777128, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(4⁷², 8388600, F₄, 8) (dual of [8388600, 8388528, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁷², large, F₄, 8) (dual of [large, large−72, 9]-code), using
        the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(68, 68+8, large)-Net over F₄ — Digital

Digital (68, 76, large)-net over F₄, using

4⁴ times duplication [i] based on digital (64, 72, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷², large, F₄, 8) (dual of [large, large−72, 9]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(68, 68+8, large)-Net in Base 4 — Upper bound on s

There is no (68, 76, large)-net in base 4, because

6 times m-reduction [i] would yield (68, 70, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]