MinT - Best Known (118−32, 118, s)-Nets in Base 4

Best Known (118−32, 118, s)-Nets in Base 4

Digital (86, 118, 531)-net over F₄, using

4¹ times duplication [i] based on digital (85, 117, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 39, 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]

Digital (86, 118, 826)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁸, 826, F₄, 32) (dual of [826, 708, 33]-code), using
- 707 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0) [i] based on linear OA(4³², 33, F₄, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,4)), using
  - dual of repetition code with length 33 [i]

There is no (86, 118, 62449)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 110447 528630 186790 977818 412167 986500 292320 993900 368693 744105 110941 867683 > 4¹¹⁸ [i]