MinT - Best Known (85, 85+36, s)-Nets in Base 4

Best Known (85, 85+36, s)-Nets in Base 4

Digital (85, 121, 312)-net over F₄, using

2 times m-reduction [i] based on digital (85, 123, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 41, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

Digital (85, 121, 577)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²¹, 577, F₄, 36) (dual of [577, 456, 37]-code), using
- 455 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (10, 5, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 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, 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, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 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, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0) [i] based on linear OA(4³⁶, 37, F₄, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,4)), using
  - dual of repetition code with length 37 [i]

There is no (85, 121, 28053)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 071440 968745 853803 199992 357446 036124 683448 184090 978942 190437 718798 065550 > 4¹²¹ [i]