MinT - Best Known (39, 39+20, s)-Nets in Base 4

Best Known (39, 39+20, s)-Nets in Base 4

(39, 39+20, 130)-Net over F₄ — Constructive and digital

Digital (39, 59, 130)-net over F₄, using

7 times m-reduction [i] based on digital (39, 66, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 33, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(39, 39+20, 207)-Net over F₄ — Digital

Digital (39, 59, 207)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁹, 207, F₄, 20) (dual of [207, 148, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁹, 255, F₄, 20) (dual of [255, 196, 21]-code), using
  - the primitive BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â {67,68,…,86}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(39, 39+20, 5374)-Net in Base 4 — Upper bound on s

There is no (39, 59, 5375)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 332319 045679 585321 309868 201076 902851 > 4⁵⁹ [i]