MinT - Best Known (51−14, 51, s)-Nets in Base 4

Best Known (51−14, 51, s)-Nets in Base 4

(51−14, 51, 312)-Net over F₄ — Constructive and digital

Digital (37, 51, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 17, 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]

(51−14, 51, 387)-Net in Base 4 — Constructive

(37, 51, 387)-net in base 4, using

trace code for nets [i] based on (3, 17, 129)-net in base 64, using
- 4 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
  - base change [i] based on digital (0, 18, 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]

(51−14, 51, 560)-Net over F₄ — Digital

Digital (37, 51, 560)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵¹, 560, F₄, 14) (dual of [560, 509, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵¹, 1023, F₄, 14) (dual of [1023, 972, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(51−14, 51, 27425)-Net in Base 4 — Upper bound on s

There is no (37, 51, 27426)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 071619 038850 611814 984933 869264 > 4⁵¹ [i]