MinT - Best Known (45, 45+26, s)-Nets in Base 8

Best Known (45, 45+26, s)-Nets in Base 8

(45, 45+26, 354)-Net over F₈ — Constructive and digital

Digital (45, 71, 354)-net over F₈, using

5 times m-reduction [i] based on digital (45, 76, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 38, 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]

(45, 45+26, 514)-Net in Base 8 — Constructive

(45, 71, 514)-net in base 8, using

1 times m-reduction [i] based on (45, 72, 514)-net in base 8, using
- base change [i] based on digital (27, 54, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(45, 45+26, 557)-Net over F₈ — Digital

Digital (45, 71, 557)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷¹, 557, F₈, 26) (dual of [557, 486, 27]-code), using
- 38 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 10 times 0, 1, 22 times 0) [i] based on linear OA(8⁶⁷, 515, F₈, 26) (dual of [515, 448, 27]-code), using
  - constructionÂ X applied to C_e(25) âŠ‚ C_e(24) [i] based on
    1. linear OA(8⁶⁷, 512, F₈, 26) (dual of [512, 445, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    2. linear OA(8⁶⁴, 512, F₈, 25) (dual of [512, 448, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    3. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 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]

(45, 45+26, 69265)-Net in Base 8 — Upper bound on s

There is no (45, 71, 69266)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 245905 987409 885900 192062 103096 717337 167027 550586 869171 820884 > 8⁷¹ [i]