MinT - Best Known (58, 81, s)-Nets in Base 8

Best Known (58, 81, s)-Nets in Base 8

(58, 81, 419)-Net over F₈ — Constructive and digital

Digital (58, 81, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 21, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(10,64) in PG(20,8)) for nets [i] based on digital (0, 11, 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 (37, 60, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 30, 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]

(58, 81, 576)-Net in Base 8 — Constructive

(58, 81, 576)-net in base 8, using

t-expansion [i] based on (57, 81, 576)-net in base 8, using
- 3 times m-reduction [i] based on (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]

(58, 81, 3405)-Net over F₈ — Digital

Digital (58, 81, 3405)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸¹, 3405, F₈, 23) (dual of [3405, 3324, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸¹, 4096, F₈, 23) (dual of [4096, 4015, 24]-code), using
  - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(58, 81, 2593236)-Net in Base 8 — Upper bound on s

There is no (58, 81, 2593237)-net in base 8, because

1 times m-reduction [i] would yield (58, 80, 2593237)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 766847 766093 433936 325468 865129 578325 445595 112075 490885 729572 803985 007080 > 8⁸⁰ [i]