MinT - Best Known (82, 115, s)-Nets in Base 8

Best Known (82, 115, s)-Nets in Base 8

(82, 115, 514)-Net over F₈ — Constructive and digital

Digital (82, 115, 514)-net over F₈, using

8¹ times duplication [i] based on digital (81, 114, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 34, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 17, 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 (47, 80, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 40, 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]

(82, 115, 590)-Net in Base 8 — Constructive

(82, 115, 590)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. (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]

(82, 115, 3697)-Net over F₈ — Digital

Digital (82, 115, 3697)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁵, 3697, F₈, 33) (dual of [3697, 3582, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹¹⁵, 4105, F₈, 33) (dual of [4105, 3990, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(29) [i] based on
    1. linear OA(8¹¹³, 4096, F₈, 33) (dual of [4096, 3983, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(8¹⁰⁵, 4096, F₈, 30) (dual of [4096, 3991, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    3. linear OA(8², 9, F₈, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
      - extended Reedâ€“Solomon code RS_e(7,8) [i]
      - Hamming code H(2,8) [i]

(82, 115, 2642136)-Net in Base 8 — Upper bound on s

There is no (82, 115, 2642137)-net in base 8, because

1 times m-reduction [i] would yield (82, 114, 2642137)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8 959011 799162 336037 184422 540350 454326 141997 463662 261227 117094 060765 280417 772998 417096 587448 391278 118715 > 8¹¹⁴ [i]