MinT - Best Known (65, 65+25, s)-Nets in Base 8

Best Known (65, 65+25, s)-Nets in Base 8

(65, 65+25, 514)-Net over F₈ — Constructive and digital

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]

(65, 65+25, 576)-Net in Base 8 — Constructive

(65, 90, 576)-net in base 8, using

8 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
- trace code for nets [i] based on (16, 49, 288)-net in base 64, using
  - base change [i] based on digital (9, 42, 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]

(65, 65+25, 4132)-Net over F₈ — Digital

Digital (65, 90, 4132)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁰, 4132, F₈, 25) (dual of [4132, 4042, 26]-code), using
- 31 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 1, 7 times 0, 1, 20 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]

(65, 65+25, 3768599)-Net in Base 8 — Upper bound on s

There is no (65, 90, 3768600)-net in base 8, because

1 times m-reduction [i] would yield (65, 89, 3768600)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 142840 769452 193242 816603 759632 299426 776788 699246 538932 104565 668523 518630 891246 > 8⁸⁹ [i]