MinT - Best Known (66, 103, s)-Nets in Base 8

Best Known (66, 103, s)-Nets in Base 8

Digital (66, 103, 354)-net over F₈, using

15 times m-reduction [i] based on digital (66, 118, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 59, 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]

(66, 103, 516)-net in base 8, using

1 times m-reduction [i] based on (66, 104, 516)-net in base 8, using
- base change [i] based on digital (40, 78, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 39, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

Digital (66, 103, 801)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰³, 801, F₈, 37) (dual of [801, 698, 38]-code), using
- 697 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 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, 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, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0) [i] based on linear OA(8³⁷, 38, F₈, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,8)), using
  - dual of repetition code with length 38 [i]

There is no (66, 103, 141420)-net in base 8, because

1 times m-reduction [i] would yield (66, 102, 141420)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 130 385906 532672 934453 199205 487841 870182 971442 952153 388839 361433 026869 903062 434246 382070 623930 > 8¹⁰² [i]