MinT - Best Known (77, 77+29, s)-Nets in Base 4

Best Known (77, 77+29, s)-Nets in Base 4

(77, 77+29, 384)-Net over F₄ — Constructive and digital

Digital (77, 106, 384)-net over F₄, using

2 times m-reduction [i] based on digital (77, 108, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 36, 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]

(77, 77+29, 450)-Net in Base 4 — Constructive

(77, 106, 450)-net in base 4, using

4¹ times duplication [i] based on (76, 105, 450)-net in base 4, using
- trace code for nets [i] based on (6, 35, 150)-net in base 64, using
  - base change [i] based on digital (1, 30, 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]

(77, 77+29, 779)-Net over F₄ — Digital

Digital (77, 106, 779)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁶, 779, F₄, 29) (dual of [779, 673, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰⁶, 1023, F₄, 29) (dual of [1023, 917, 30]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(77, 77+29, 66025)-Net in Base 4 — Upper bound on s

There is no (77, 106, 66026)-net in base 4, because

1 times m-reduction [i] would yield (77, 105, 66026)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1645 611022 661624 040121 719096 515404 461564 110079 263562 318558 141616 > 4¹⁰⁵ [i]