MinT - Best Known (81−21, 81, s)-Nets in Base 4

Best Known (81−21, 81, s)-Nets in Base 4

(81−21, 81, 384)-Net over F₄ — Constructive and digital

Digital (60, 81, 384)-net over F₄, using

t-expansion [i] based on digital (59, 81, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 27, 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]

(81−21, 81, 450)-Net in Base 4 — Constructive

(60, 81, 450)-net in base 4, using

t-expansion [i] based on (59, 81, 450)-net in base 4, using
- trace code for nets [i] based on (5, 27, 150)-net in base 64, using
  - 1 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
    - base change [i] based on digital (1, 24, 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]

(81−21, 81, 892)-Net over F₄ — Digital

Digital (60, 81, 892)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸¹, 892, F₄, 21) (dual of [892, 811, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸¹, 1023, F₄, 21) (dual of [1023, 942, 22]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {−6,−5,…,14}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(81−21, 81, 98923)-Net in Base 4 — Upper bound on s

There is no (60, 81, 98924)-net in base 4, because

1 times m-reduction [i] would yield (60, 80, 98924)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 461537 878693 244653 273282 342669 169471 890113 408951 > 4⁸⁰ [i]