MinT - Best Known (66−25, 66, s)-Nets in Base 8

Best Known (66−25, 66, s)-Nets in Base 8

(66−25, 66, 354)-Net over F₈ — Constructive and digital

Digital (41, 66, 354)-net over F₈, using

2 times m-reduction [i] based on digital (41, 68, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 34, 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]

(66−25, 66, 384)-Net in Base 8 — Constructive

(41, 66, 384)-net in base 8, using

trace code for nets [i] based on (8, 33, 192)-net in base 64, using
- 2 times m-reduction [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]

(66−25, 66, 468)-Net over F₈ — Digital

Digital (41, 66, 468)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁶, 468, F₈, 25) (dual of [468, 402, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶⁶, 511, F₈, 25) (dual of [511, 445, 26]-code), using
  - 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]

(66−25, 66, 58877)-Net in Base 8 — Upper bound on s

There is no (41, 66, 58878)-net in base 8, because

1 times m-reduction [i] would yield (41, 65, 58878)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 50224 382819 496700 559300 537300 914108 970726 225827 661071 709883 > 8⁶⁵ [i]