MinT - Best Known (154−43, 154, s)-Nets in Base 4

Best Known (154−43, 154, s)-Nets in Base 4

Digital (111, 154, 531)-net over F₄, using

2 times m-reduction [i] based on digital (111, 156, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 52, 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 (111, 154, 909)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵⁴, 909, F₄, 43) (dual of [909, 755, 44]-code), using
- 754 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (12, 5, 3, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 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, 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, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 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, 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, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 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, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0) [i] based on linear OA(4⁴³, 44, F₄, 43) (dual of [44, 1, 44]-code or 44-arc in PG(42,4)), using
  - dual of repetition code with length 44 [i]

There is no (111, 154, 70422)-net in base 4, because

1 times m-reduction [i] would yield (111, 153, 70422)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 400437 970415 383370 392250 045827 079153 230314 354984 212164 050421 077330 801334 487345 741749 747527 > 4¹⁵³ [i]