MinT - Best Known (35−10, 35, s)-Nets in Base 4

Best Known (35−10, 35, s)-Nets in Base 4

(35−10, 35, 240)-Net over F₄ — Constructive and digital

Digital (25, 35, 240)-net over F₄, using

1 times m-reduction [i] based on digital (25, 36, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 12, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(35−10, 35, 346)-Net over F₄ — Digital

Digital (25, 35, 346)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁵, 346, F₄, 10) (dual of [346, 311, 11]-code), using
- constructionÂ X applied to C([106,115]) âŠ‚ C([107,115]) [i] based on
  1. linear OA(4³⁵, 341, F₄, 10) (dual of [341, 306, 11]-code), using the BCH-code C(I) with length 341Â | 4⁵âˆ’1, defining interval IÂ =Â {106,107,…,115}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
  2. linear OA(4³⁰, 341, F₄, 9) (dual of [341, 311, 10]-code), using the BCH-code C(I) with length 341Â | 4⁵âˆ’1, defining interval IÂ =Â {107,108,…,115}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
  3. linear OA(4⁰, 5, F₄, 0) (dual of [5, 5, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(35−10, 35, 14224)-Net in Base 4 — Upper bound on s

There is no (25, 35, 14225)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1180 989389 586469 258336 > 4³⁵ [i]