MinT - Best Known (68, 68+29, s)-Nets in Base 4

Best Known (68, 68+29, s)-Nets in Base 4

Digital (68, 97, 312)-net over F₄, using

4¹ times duplication [i] based on digital (67, 96, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 32, 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 (68, 97, 473)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁹⁷, 473, F₄, 29) (dual of [473, 376, 30]-code), using
- 375 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 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, 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, 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, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0) [i] based on linear OA(4²⁹, 30, F₄, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,4)), using
  - dual of repetition code with length 30 [i]

There is no (68, 97, 27076)-net in base 4, because

1 times m-reduction [i] would yield (68, 96, 27076)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6280 278904 057411 644176 337356 127369 211839 610842 658968 436336 > 4⁹⁶ [i]