MinT - Best Known (66, 66+25, s)-Nets in Base 8

Best Known (66, 66+25, s)-Nets in Base 8

(66, 66+25, 514)-Net over F₈ — Constructive and digital

Digital (66, 91, 514)-net over F₈, using

8¹ times duplication [i] based on digital (65, 90, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 26, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 13, 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 (39, 64, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 32, 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]

(66, 66+25, 579)-Net in Base 8 — Constructive

(66, 91, 579)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (11, 23, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(11,64) in PG(22,8)) for nets [i] based on digital (0, 12, 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. (43, 68, 514)-net in base 8, using
  - base change [i] based on digital (26, 51, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (26, 52, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 26, 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]

(66, 66+25, 4188)-Net over F₈ — Digital

Digital (66, 91, 4188)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹¹, 4188, F₈, 25) (dual of [4188, 4097, 26]-code), using
- 86 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 7 times 0, 1, 20 times 0, 1, 54 times 0) [i] based on linear OA(8⁸⁵, 4096, F₈, 25) (dual of [4096, 4011, 26]-code), using
  - an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(66, 66+25, 4481646)-Net in Base 8 — Upper bound on s

There is no (66, 91, 4481647)-net in base 8, because

1 times m-reduction [i] would yield (66, 90, 4481647)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1897 141209 382116 734704 713890 351482 115716 059511 284626 260765 528345 362129 275778 921871 > 8⁹⁰ [i]