MinT - Best Known (44, 44+21, s)-Nets in Base 8

Best Known (44, 44+21, s)-Nets in Base 8

Digital (44, 65, 354)-net over F₈, using

9 times m-reduction [i] based on digital (44, 74, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 37, 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]

(44, 65, 520)-net in base 8, using

8¹ times duplication [i] based on (43, 64, 520)-net in base 8, using
- base change [i] based on digital (27, 48, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 24, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

Digital (44, 65, 1032)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁵, 1032, F₈, 21) (dual of [1032, 967, 22]-code), using
- 966 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 17 times 0, 1, 20 times 0, 1, 21 times 0, 1, 24 times 0, 1, 27 times 0, 1, 30 times 0, 1, 34 times 0, 1, 38 times 0, 1, 42 times 0, 1, 46 times 0, 1, 52 times 0, 1, 58 times 0, 1, 65 times 0, 1, 72 times 0, 1, 80 times 0, 1, 89 times 0, 1, 99 times 0) [i] based on linear OA(8²¹, 22, F₈, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,8)), using
  - dual of repetition code with length 22 [i]

There is no (44, 65, 389626)-net in base 8, because

1 times m-reduction [i] would yield (44, 64, 389626)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6277 153490 257261 956728 278731 549135 336098 885186 892187 948824 > 8⁶⁴ [i]