MinT - Best Known (85−22, 85, s)-Nets in Base 4

Best Known (85−22, 85, s)-Nets in Base 4

(85−22, 85, 384)-Net over F₄ — Constructive and digital

Digital (63, 85, 384)-net over F₄, using

2 times m-reduction [i] based on digital (63, 87, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 29, 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]

(85−22, 85, 450)-Net in Base 4 — Constructive

(63, 85, 450)-net in base 4, using

4¹ times duplication [i] based on (62, 84, 450)-net in base 4, using
- t-expansion [i] based on (61, 84, 450)-net in base 4, using
  - trace code for nets [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]

(85−22, 85, 920)-Net over F₄ — Digital

Digital (63, 85, 920)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁵, 920, F₄, 22) (dual of [920, 835, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸⁵, 1023, F₄, 22) (dual of [1023, 938, 23]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(85−22, 85, 73472)-Net in Base 4 — Upper bound on s

There is no (63, 85, 73473)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1496 742316 847000 059361 618697 708213 485965 498522 113280 > 4⁸⁵ [i]