MinT - Best Known (71, 71+27, s)-Nets in Base 8

Best Known (71, 71+27, s)-Nets in Base 8

(71, 71+27, 514)-Net over F₈ — Constructive and digital

Digital (71, 98, 514)-net over F₈, using

8² 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]

(71, 71+27, 644)-Net in Base 8 — Constructive

(71, 98, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (13, 26, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 13, 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. (45, 72, 514)-net in base 8, using
  - base change [i] based on digital (27, 54, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(71, 71+27, 4187)-Net over F₈ — Digital

Digital (71, 98, 4187)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁸, 4187, F₈, 27) (dual of [4187, 4089, 28]-code), using
- 82 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 1, 6 times 0, 1, 19 times 0, 1, 52 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]

(71, 71+27, 4433513)-Net in Base 8 — Upper bound on s

There is no (71, 98, 4433514)-net in base 8, because

1 times m-reduction [i] would yield (71, 97, 4433514)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3978 591074 875835 858503 521879 177890 071405 777514 076604 962658 267788 538848 709374 292651 375080 > 8⁹⁷ [i]