MinT - Best Known (135−38, 135, s)-Nets in Base 4

Best Known (135−38, 135, s)-Nets in Base 4

Digital (97, 135, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 45, 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 (97, 135, 787)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁵, 787, F₄, 38) (dual of [787, 652, 39]-code), using
- 651 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (11, 5, 3, 2, 1, 1, 2, 1, 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, 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, 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, 6 times 0, 1, 6 times 0, 1, 7 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, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0) [i] based on linear OA(4³⁸, 39, F₄, 38) (dual of [39, 1, 39]-code or 39-arc in PG(37,4)), using
  - dual of repetition code with length 39 [i]

There is no (97, 135, 50091)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1897 443097 968981 871505 133277 486540 565931 506040 702884 410952 536981 084097 541766 118152 > 4¹³⁵ [i]