MinT - Best Known (72, 72+29, s)-Nets in Base 8

Best Known (72, 72+29, s)-Nets in Base 8

(72, 72+29, 484)-Net over F₈ — Constructive and digital

Digital (72, 101, 484)-net over F₈, using

8¹ times duplication [i] based on digital (71, 100, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 28, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 14, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(72, 72+29, 576)-Net in Base 8 — Constructive

(72, 101, 576)-net in base 8, using

9 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(72, 72+29, 3436)-Net over F₈ — Digital

Digital (72, 101, 3436)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰¹, 3436, F₈, 29) (dual of [3436, 3335, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(72, 72+29, 2437816)-Net in Base 8 — Upper bound on s

There is no (72, 101, 2437817)-net in base 8, because

1 times m-reduction [i] would yield (72, 100, 2437817)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 037045 948874 112941 579784 735141 488663 375636 538956 874823 975547 005690 736085 748975 551365 544664 > 8¹⁰⁰ [i]