MinT - Best Known (70−25, 70, s)-Nets in Base 8

Best Known (70−25, 70, s)-Nets in Base 8

Digital (45, 70, 354)-net over F₈, using

6 times m-reduction [i] based on digital (45, 76, 354)-net over F₈, 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]

(45, 70, 516)-net in base 8, using

trace code for nets [i] based on (10, 35, 258)-net in base 64, using
- 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
  - base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

Digital (45, 70, 617)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁰, 617, F₈, 25) (dual of [617, 547, 26]-code), using
- 99 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 6 times 0, 1, 15 times 0, 1, 29 times 0, 1, 43 times 0) [i] based on linear OA(8⁶⁴, 512, F₈, 25) (dual of [512, 448, 26]-code), using
  - an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

There is no (45, 70, 117762)-net in base 8, because

1 times m-reduction [i] would yield (45, 69, 117762)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 205 693629 320177 883599 535935 503107 926277 624390 148718 876933 075712 > 8⁶⁹ [i]