MinT - Best Known (26, 41, s)-Nets in Base 8

Best Known (26, 41, s)-Nets in Base 8

(26, 41, 256)-Net over F₈ — Constructive and digital

Digital (26, 41, 256)-net over F₈, using

1 times m-reduction [i] based on digital (26, 42, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 21, 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]

(26, 41, 479)-Net over F₈ — Digital

Digital (26, 41, 479)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴¹, 479, F₈, 15) (dual of [479, 438, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴¹, 519, F₈, 15) (dual of [519, 478, 16]-code), using
  - constructionÂ X applied to C_e(14) âŠ‚ C_e(12) [i] based on
    1. linear OA(8⁴⁰, 512, F₈, 15) (dual of [512, 472, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
    2. linear OA(8³⁴, 512, F₈, 13) (dual of [512, 478, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(8¹, 7, F₈, 1) (dual of [7, 6, 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]

(26, 41, 514)-Net in Base 8 — Constructive

(26, 41, 514)-net in base 8, using

8¹ times duplication [i] based on (25, 40, 514)-net in base 8, using
- base change [i] based on digital (15, 30, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 15, 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]

(26, 41, 69873)-Net in Base 8 — Upper bound on s

There is no (26, 41, 69874)-net in base 8, because

1 times m-reduction [i] would yield (26, 40, 69874)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 329348 975352 540953 207143 903640 247352 > 8⁴⁰ [i]