MinT - Best Known (240−180, 240, s)-Nets in Base 4

Best Known (240−180, 240, s)-Nets in Base 4

(240−180, 240, 66)-Net over F₄ — Constructive and digital

Digital (60, 240, 66)-net over F₄, using

t-expansion [i] based on digital (49, 240, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(240−180, 240, 91)-Net over F₄ — Digital

Digital (60, 240, 91)-net over F₄, using

t-expansion [i] based on digital (50, 240, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(240−180, 240, 251)-Net over F₄ — Upper bound on s (digital)

There is no digital (60, 240, 252)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁰, 252, F₄, 180) (dual of [252, 12, 181]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(4²³⁹, 246, F₄, 180) (dual of [246, 7, 181]-code), but
    - residual code [i] would yield linear OA(4⁵⁹, 65, F₄, 45) (dual of [65, 6, 46]-code), but
      - 1 times truncation [i] would yield linear OA(4⁵⁸, 64, F₄, 44) (dual of [64, 6, 45]-code), but
        residual code [i] would yield linear OA(4¹⁴, 19, F₄, 11) (dual of [19, 5, 12]-code), but
        1 times truncation [i] would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
        â€œLizâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(4¹², 252, S₄, 6), but
    - discarding factors would yield OA(4¹², 156, S₄, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 16 866019 > 4¹² [i]

(240−180, 240, 390)-Net in Base 4 — Upper bound on s

There is no (60, 240, 391)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 345390 416691 786881 934492 291322 306282 008467 896870 263413 306916 867408 757116 375789 626040 491020 836325 313345 805694 609119 493345 273387 836537 993496 341328 > 4²⁴⁰ [i]