MinT - Best Known (71−27, 71, s)-Nets in Base 8

Best Known (71−27, 71, s)-Nets in Base 8

(71−27, 71, 354)-Net over F₈ — Constructive and digital

Digital (44, 71, 354)-net over F₈, using

3 times m-reduction [i] based on digital (44, 74, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 37, 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]

(71−27, 71, 384)-Net in Base 8 — Constructive

(44, 71, 384)-net in base 8, using

8¹ times duplication [i] based on (43, 70, 384)-net in base 8, using
- trace code for nets [i] based on (8, 35, 192)-net in base 64, using
  - base change [i] based on digital (3, 30, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(71−27, 71, 478)-Net over F₈ — Digital

Digital (44, 71, 478)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷¹, 478, F₈, 27) (dual of [478, 407, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷¹, 519, F₈, 27) (dual of [519, 448, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(24) [i] based on
    1. linear OA(8⁷⁰, 512, F₈, 27) (dual of [512, 442, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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(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]

(71−27, 71, 59025)-Net in Base 8 — Upper bound on s

There is no (44, 71, 59026)-net in base 8, because

1 times m-reduction [i] would yield (44, 70, 59026)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1645 543062 097734 350805 755329 296524 851445 663098 549178 022168 693588 > 8⁷⁰ [i]