MinT - Best Known (113−32, 113, s)-Nets in Base 8

Best Known (113−32, 113, s)-Nets in Base 8

(113−32, 113, 514)-Net over F₈ — Constructive and digital

Digital (81, 113, 514)-net over F₈, using

1 times m-reduction [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]

(113−32, 113, 590)-Net in Base 8 — Constructive

(81, 113, 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. (64, 96, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 48, 288)-net in base 64, using
    - 1 times m-reduction [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]

(113−32, 113, 4030)-Net over F₈ — Digital

Digital (81, 113, 4030)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹³, 4030, F₈, 32) (dual of [4030, 3917, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹¹³, 4100, F₈, 32) (dual of [4100, 3987, 33]-code), using
  - 1 times truncation [i] based on linear OA(8¹¹⁴, 4101, F₈, 33) (dual of [4101, 3987, 34]-code), using
    - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [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(8¹⁰⁹, 4096, F₈, 31) (dual of [4096, 3987, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      3. linear OA(8¹, 5, F₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(113−32, 113, 2320127)-Net in Base 8 — Upper bound on s

There is no (81, 113, 2320128)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 119874 818305 816154 069300 327863 799231 228634 546003 228999 839191 482303 078077 906322 004822 442807 526947 086769 > 8¹¹³ [i]