MinT - Best Known (34−9, 34, s)-Nets in Base 4

Best Known (34−9, 34, s)-Nets in Base 4

(34−9, 34, 514)-Net over F₄ — Constructive and digital

Digital (25, 34, 514)-net over F₄, using

trace code for nets [i] based on digital (8, 17, 257)-net over F₁₆, using
- base reduction for projective spaces (embedding PG(8,256) in PG(16,16)) for nets [i] based on digital (0, 9, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]

(34−9, 34, 772)-Net over F₄ — Digital

Digital (25, 34, 772)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁴, 772, F₄, 9) (dual of [772, 738, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁴, 1031, F₄, 9) (dual of [1031, 997, 10]-code), using
  - constructionÂ X applied to C([0,4]) âŠ‚ C([0,2]) [i] based on
    1. linear OA(4³¹, 1025, F₄, 9) (dual of [1025, 994, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OA(4²¹, 1025, F₄, 5) (dual of [1025, 1004, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 4¹⁰âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−4,−2,0,2,4}|+1Â =Â 6 (BCH-bound) [i]
    3. linear OA(4³, 6, F₄, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
      - hyperoval in PG(2,Â 4) [i]
      - the Hexacode [i]

(34−9, 34, 68376)-Net in Base 4 — Upper bound on s

There is no (25, 34, 68377)-net in base 4, because

1 times m-reduction [i] would yield (25, 33, 68377)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 73 788032 037158 226274 > 4³³ [i]