MinT - Best Known (132−37, 132, s)-Nets in Base 4

Best Known (132−37, 132, s)-Nets in Base 4

Digital (95, 132, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 44, 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 (95, 132, 786)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³², 786, F₄, 37) (dual of [786, 654, 38]-code), using
- 653 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 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, 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, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 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, 9 times 0, 1, 9 times 0, 1, 10 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, 14 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, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0) [i] based on linear OA(4³⁷, 38, F₄, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,4)), using
  - dual of repetition code with length 38 [i]

There is no (95, 132, 60613)-net in base 4, because

1 times m-reduction [i] would yield (95, 131, 60613)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 411315 075623 867218 576809 036666 972772 264653 635991 071451 939405 165026 787152 357880 > 4¹³¹ [i]