MinT - Best Known (61−16, 61, s)-Nets in Base 4

Best Known (61−16, 61, s)-Nets in Base 4

(61−16, 61, 312)-Net over F₄ — Constructive and digital

Digital (45, 61, 312)-net over F₄, using

2 times m-reduction [i] based on digital (45, 63, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 21, 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]

(61−16, 61, 450)-Net in Base 4 — Constructive

(45, 61, 450)-net in base 4, using

4¹ times duplication [i] based on (44, 60, 450)-net in base 4, using
- trace code for nets [i] based on (4, 20, 150)-net in base 64, using
  - 1 times m-reduction [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]

(61−16, 61, 756)-Net over F₄ — Digital

Digital (45, 61, 756)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶¹, 756, F₄, 16) (dual of [756, 695, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶¹, 1023, F₄, 16) (dual of [1023, 962, 17]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {−6,−5,…,9}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(61−16, 61, 48890)-Net in Base 4 — Upper bound on s

There is no (45, 61, 48891)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 317641 445048 335723 941586 762741 027847 > 4⁶¹ [i]