MinT - Best Known (33, 45, s)-Nets in Base 4

Best Known (33, 45, s)-Nets in Base 4

(33, 45, 312)-Net over F₄ — Constructive and digital

Digital (33, 45, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 15, 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]

(33, 45, 387)-Net in Base 4 — Constructive

(33, 45, 387)-net in base 4, using

4³ times duplication [i] based on (30, 42, 387)-net in base 4, using
- trace code for nets [i] based on (2, 14, 129)-net in base 64, using
  - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(33, 45, 666)-Net over F₄ — Digital

Digital (33, 45, 666)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁵, 666, F₄, 12) (dual of [666, 621, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴⁵, 1023, F₄, 12) (dual of [1023, 978, 13]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(33, 45, 32695)-Net in Base 4 — Upper bound on s

There is no (33, 45, 32696)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1238 002071 225001 241733 310699 > 4⁴⁵ [i]