MinT - Best Known (53, 86, s)-Nets in Base 8

Best Known (53, 86, s)-Nets in Base 8

(53, 86, 354)-Net over F₈ — Constructive and digital

Digital (53, 86, 354)-net over F₈, using

6 times m-reduction [i] based on digital (53, 92, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 46, 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]

(53, 86, 384)-Net in Base 8 — Constructive

(53, 86, 384)-net in base 8, using

8² times duplication [i] based on (51, 84, 384)-net in base 8, using
- trace code for nets [i] based on (9, 42, 192)-net in base 64, using
  - base change [i] based on digital (3, 36, 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]

(53, 86, 514)-Net over F₈ — Digital

Digital (53, 86, 514)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁶, 514, F₈, 33) (dual of [514, 428, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁶, 516, F₈, 33) (dual of [516, 430, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
    1. linear OA(8⁸⁵, 512, F₈, 33) (dual of [512, 427, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(8⁸², 512, F₈, 31) (dual of [512, 430, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(8¹, 4, F₈, 1) (dual of [4, 3, 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]

(53, 86, 60958)-Net in Base 8 — Upper bound on s

There is no (53, 86, 60959)-net in base 8, because

1 times m-reduction [i] would yield (53, 85, 60959)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57898 820646 089106 773888 001988 783804 626428 202921 700035 565697 349667 464108 384819 > 8⁸⁵ [i]