MinT - Best Known (26, 42, s)-Nets in Base 8

Best Known (26, 42, s)-Nets in Base 8

(26, 42, 256)-Net over F₈ — Constructive and digital

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, 42, 300)-Net in Base 8 — Constructive

(26, 42, 300)-net in base 8, using

t-expansion [i] based on (25, 42, 300)-net in base 8, using
- trace code for nets [i] based on (4, 21, 150)-net in base 64, using
  - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(26, 42, 374)-Net over F₈ — Digital

Digital (26, 42, 374)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴², 374, F₈, 16) (dual of [374, 332, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴², 511, F₈, 16) (dual of [511, 469, 17]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(26, 42, 29631)-Net in Base 8 — Upper bound on s

There is no (26, 42, 29632)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 85 093389 120195 041495 483069 207457 723545 > 8⁴² [i]