MinT - Best Known (75, 75+27, s)-Nets in Base 4

Best Known (75, 75+27, s)-Nets in Base 4

(75, 75+27, 531)-Net over F₄ — Constructive and digital

Digital (75, 102, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 34, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(75, 75+27, 899)-Net over F₄ — Digital

Digital (75, 102, 899)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰², 899, F₄, 27) (dual of [899, 797, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰², 1036, F₄, 27) (dual of [1036, 934, 28]-code), using
  - constructionÂ X applied to C([0,13]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(4¹⁰¹, 1025, F₄, 27) (dual of [1025, 924, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    2. linear OA(4⁹¹, 1025, F₄, 25) (dual of [1025, 934, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    3. linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(75, 75+27, 89903)-Net in Base 4 — Upper bound on s

There is no (75, 102, 89904)-net in base 4, because

1 times m-reduction [i] would yield (75, 101, 89904)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 427943 299843 177404 112345 221604 449460 846162 447758 350485 025827 > 4¹⁰¹ [i]