MinT - Best Known (125−33, 125, s)-Nets in Base 4

Best Known (125−33, 125, s)-Nets in Base 4

(125−33, 125, 531)-Net over F₄ — Constructive and digital

Digital (92, 125, 531)-net over F₄, using

t-expansion [i] based on digital (91, 125, 531)-net over F₄, using
- 1 times m-reduction [i] based on digital (91, 126, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 42, 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]

(125−33, 125, 1036)-Net over F₄ — Digital

Digital (92, 125, 1036)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁵, 1036, F₄, 33) (dual of [1036, 911, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²⁵, 1039, F₄, 33) (dual of [1039, 914, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,14]) [i] based on
    1. linear OA(4¹²¹, 1025, F₄, 33) (dual of [1025, 904, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹¹¹, 1025, F₄, 29) (dual of [1025, 914, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(4⁴, 14, F₄, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]

(125−33, 125, 105033)-Net in Base 4 — Upper bound on s

There is no (92, 125, 105034)-net in base 4, because

1 times m-reduction [i] would yield (92, 124, 105034)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 452 322177 225512 573644 311743 602198 008226 492892 047755 760539 846225 196907 789524 > 4¹²⁴ [i]