MinT - Best Known (105−32, 105, s)-Nets in Base 4

Best Known (105−32, 105, s)-Nets in Base 4

Digital (73, 105, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 35, 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 (73, 105, 469)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁵, 469, F₄, 32) (dual of [469, 364, 33]-code), using
- 363 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) [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 (73, 105, 20238)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 908880 619890 392491 374456 442057 559636 733984 260038 137450 315120 > 4¹⁰⁵ [i]