MinT - Best Known (73, 102, s)-Nets in Base 8

Best Known (73, 102, s)-Nets in Base 8

(73, 102, 514)-Net over F₈ — Constructive and digital

Digital (73, 102, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (16, 30, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 15, 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 (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(73, 102, 576)-Net in Base 8 — Constructive

(73, 102, 576)-net in base 8, using

10 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(73, 102, 3712)-Net over F₈ — Digital

Digital (73, 102, 3712)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰², 3712, F₈, 29) (dual of [3712, 3610, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁰², 4105, F₈, 29) (dual of [4105, 4003, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(26) [i] based on
    1. linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(8⁹³, 4096, F₈, 27) (dual of [4096, 4003, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 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]

(73, 102, 2828183)-Net in Base 8 — Upper bound on s

There is no (73, 102, 2828184)-net in base 8, because

1 times m-reduction [i] would yield (73, 101, 2828184)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 16 296327 185438 342120 438947 501825 673614 364822 355928 079336 671288 883371 621231 246194 295561 122678 > 8¹⁰¹ [i]