MinT - Best Known (24−9, 24, s)-Nets in Base 8

Best Known (24−9, 24, s)-Nets in Base 8

(24−9, 24, 208)-Net over F₈ — Constructive and digital

Digital (15, 24, 208)-net over F₈, using

trace code for nets [i] based on digital (3, 12, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(24−9, 24, 444)-Net over F₈ — Digital

Digital (15, 24, 444)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8²⁴, 444, F₈, 9) (dual of [444, 420, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8²⁴, 511, F₈, 9) (dual of [511, 487, 10]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(24−9, 24, 514)-Net in Base 8 — Constructive

(15, 24, 514)-net in base 8, using

base change [i] based on digital (9, 18, 514)-net over F₁₆, using
- trace code for nets [i] based on digital (0, 9, 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]

(24−9, 24, 49283)-Net in Base 8 — Upper bound on s

There is no (15, 24, 49284)-net in base 8, because

1 times m-reduction [i] would yield (15, 23, 49284)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 590 306717 844997 979008 > 8²³ [i]