MinT - Best Known (41−11, 41, s)-Nets in Base 4

Best Known (41−11, 41, s)-Nets in Base 4

(41−11, 41, 257)-Net over F₄ — Constructive and digital

Digital (30, 41, 257)-net over F₄, using

base reduction for projective spaces (embedding PG(10,256) in PG(40,4)) for nets [i] based on digital (0, 11, 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]

(41−11, 41, 387)-Net in Base 4 — Constructive

(30, 41, 387)-net in base 4, using

1 times m-reduction [i] based on (30, 42, 387)-net in base 4, using
- trace code for nets [i] based on (2, 14, 129)-net in base 64, using
  - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(41−11, 41, 649)-Net over F₄ — Digital

Digital (30, 41, 649)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴¹, 649, F₄, 11) (dual of [649, 608, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴¹, 1023, F₄, 11) (dual of [1023, 982, 12]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(41−11, 41, 56907)-Net in Base 4 — Upper bound on s

There is no (30, 41, 56908)-net in base 4, because

1 times m-reduction [i] would yield (30, 40, 56908)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 209014 598209 071026 634882 > 4⁴⁰ [i]