MinT - Best Known (63−26, 63, s)-Nets in Base 8

Best Known (63−26, 63, s)-Nets in Base 8

(63−26, 63, 256)-Net over F₈ — Constructive and digital

Digital (37, 63, 256)-net over F₈, using

1 times m-reduction [i] based on digital (37, 64, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 32, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(63−26, 63, 258)-Net in Base 8 — Constructive

(37, 63, 258)-net in base 8, using

1 times m-reduction [i] based on (37, 64, 258)-net in base 8, using
- trace code for nets [i] based on (5, 32, 129)-net in base 64, using
  - 3 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
    - base change [i] based on digital (0, 30, 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]

(63−26, 63, 288)-Net over F₈ — Digital

Digital (37, 63, 288)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶³, 288, F₈, 26) (dual of [288, 225, 27]-code), using
- 26 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 7 times 0, 1, 17 times 0) [i] based on linear OA(8⁶¹, 260, F₈, 26) (dual of [260, 199, 27]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(8⁶⁰, 258, F₈, 26) (dual of [258, 198, 27]-code), using
      - trace code [i] based on linear OA(64³⁰, 129, F₆₄, 26) (dual of [129, 99, 27]-code), using
        extended algebraic-geometric code AG_e(F,102P) [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(8⁶⁰, 259, F₈, 25) (dual of [259, 199, 26]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8⁶⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 786175 140070 077847 813691 509886 798809 858889 083632 283616 [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]

(63−26, 63, 19259)-Net in Base 8 — Upper bound on s

There is no (37, 63, 19260)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 784 925310 618509 665946 666252 999788 576287 159136 150293 815996 > 8⁶³ [i]