MinT - Best Known (70, 97, s)-Nets in Base 8

Best Known (70, 97, s)-Nets in Base 8

(70, 97, 514)-Net over F₈ — Constructive and digital

Digital (70, 97, 514)-net over F₈, using

8¹ times duplication [i] based on digital (69, 96, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 28, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 14, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (41, 68, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 34, 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]

(70, 97, 585)-Net in Base 8 — Constructive

(70, 97, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (57, 84, 576)-net in base 8, using
  - trace code for nets [i] based on (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 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]

(70, 97, 4133)-Net over F₈ — Digital

Digital (70, 97, 4133)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁷, 4133, F₈, 27) (dual of [4133, 4036, 28]-code), using
- 29 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 6 times 0, 1, 19 times 0) [i] based on linear OA(8⁹³, 4100, F₈, 27) (dual of [4100, 4007, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(25) [i] based on
    1. linear OA(8⁹³, 4096, F₈, 27) (dual of [4096, 4003, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. linear OA(8⁸⁹, 4096, F₈, 26) (dual of [4096, 4007, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(70, 97, 3778152)-Net in Base 8 — Upper bound on s

There is no (70, 97, 3778153)-net in base 8, because

1 times m-reduction [i] would yield (70, 96, 3778153)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 497 324643 017475 200498 097488 694668 883959 184388 066794 821098 015215 245040 670145 378882 348064 > 8⁹⁶ [i]