MinT - Best Known (45, 45+28, s)-Nets in Base 8

Best Known (45, 45+28, s)-Nets in Base 8

(45, 45+28, 354)-Net over F₈ — Constructive and digital

Digital (45, 73, 354)-net over F₈, using

3 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, 45+28, 464)-Net over F₈ — Digital

Digital (45, 73, 464)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷³, 464, F₈, 28) (dual of [464, 391, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷³, 511, F₈, 28) (dual of [511, 438, 29]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(45, 45+28, 44182)-Net in Base 8 — Upper bound on s

There is no (45, 73, 44183)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 842765 096208 863678 318841 497429 597649 432711 461353 462338 525540 365514 > 8⁷³ [i]