MinT - Best Known (51−21, 51, s)-Nets in Base 8

Best Known (51−21, 51, s)-Nets in Base 8

(51−21, 51, 208)-Net over F₈ — Constructive and digital

Digital (30, 51, 208)-net over F₈, using

3 times m-reduction [i] based on digital (30, 54, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 27, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(51−21, 51, 258)-Net in Base 8 — Constructive

(30, 51, 258)-net in base 8, using

1 times m-reduction [i] based on (30, 52, 258)-net in base 8, using
- trace code for nets [i] based on (4, 26, 129)-net in base 64, using
  - 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
    - base change [i] based on digital (0, 24, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]

(51−21, 51, 260)-Net over F₈ — Digital

Digital (30, 51, 260)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵¹, 260, F₈, 21) (dual of [260, 209, 22]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(8⁵⁰, 258, F₈, 21) (dual of [258, 208, 22]-code), using
    - trace code [i] based on linear OA(64²⁵, 129, F₆₄, 21) (dual of [129, 104, 22]-code), using
      - extended algebraic-geometric code AG_e(F,107P) [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]
  2. linear OA(8⁵⁰, 259, F₈, 20) (dual of [259, 209, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8⁵⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 318 049696 282331 499963 882546 383698 991672 378816 [i]
  3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(51−21, 51, 21193)-Net in Base 8 — Upper bound on s

There is no (30, 51, 21194)-net in base 8, because

1 times m-reduction [i] would yield (30, 50, 21194)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1427 451078 692485 029169 022485 073105 494576 227792 > 8⁵⁰ [i]