MinT - Best Known (33, 33+18, s)-Nets in Base 4

Best Known (33, 33+18, s)-Nets in Base 4

(33, 33+18, 130)-Net over F₄ — Constructive and digital

Digital (33, 51, 130)-net over F₄, using

3 times m-reduction [i] based on digital (33, 54, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 27, 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]

(33, 33+18, 162)-Net over F₄ — Digital

Digital (33, 51, 162)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵¹, 162, F₄, 18) (dual of [162, 111, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵¹, 255, F₄, 18) (dual of [255, 204, 19]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(33, 33+18, 3560)-Net in Base 4 — Upper bound on s

There is no (33, 51, 3561)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 083302 332580 970426 998552 563860 > 4⁵¹ [i]