MinT - Best Known (41, 41+21, s)-Nets in Base 4

Best Known (41, 41+21, s)-Nets in Base 4

(41, 41+21, 130)-Net over F₄ — Constructive and digital

Digital (41, 62, 130)-net over F₄, using

8 times m-reduction [i] based on digital (41, 70, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 35, 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]

(41, 41+21, 214)-Net over F₄ — Digital

Digital (41, 62, 214)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶², 214, F₄, 21) (dual of [214, 152, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁶², 255, F₄, 21) (dual of [255, 193, 22]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(41, 41+21, 7094)-Net in Base 4 — Upper bound on s

There is no (41, 62, 7095)-net in base 4, because

1 times m-reduction [i] would yield (41, 61, 7095)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 318187 931445 437255 198717 249880 947570 > 4⁶¹ [i]