MinT - Best Known (115−31, 115, s)-Nets in Base 4

Best Known (115−31, 115, s)-Nets in Base 4

Digital (84, 115, 531)-net over F₄, using

4¹ times duplication [i] based on digital (83, 114, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 38, 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 (84, 115, 831)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁵, 831, F₄, 31) (dual of [831, 716, 32]-code), using
- 715 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 2, 2, 2, 1, 1, 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, 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, 4 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, 8 times 0, 1, 9 times 0, 1, 10 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, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0) [i] based on linear OA(4³¹, 32, F₄, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,4)), using
  - dual of repetition code with length 32 [i]

There is no (84, 115, 80582)-net in base 4, because

1 times m-reduction [i] would yield (84, 114, 80582)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 431 379813 477175 770085 860026 971048 881039 051615 232129 112139 409417 427488 > 4¹¹⁴ [i]