MinT - Best Known (61, 88, s)-Nets in Base 4

Best Known (61, 88, s)-Nets in Base 4

Digital (61, 88, 240)-net over F₄, using

2 times m-reduction [i] based on digital (61, 90, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 30, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

Digital (61, 88, 397)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁸, 397, F₄, 27) (dual of [397, 309, 28]-code), using
- 308 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 4, 2, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 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, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0) [i] based on linear OA(4²⁷, 28, F₄, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,4)), using
  - dual of repetition code with length 28 [i]

There is no (61, 88, 20195)-net in base 4, because

1 times m-reduction [i] would yield (61, 87, 20195)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23952 655742 848641 766989 347049 824236 476759 416825 436400 > 4⁸⁷ [i]