MinT - Best Known (49−17, 49, s)-Nets in Base 4

Best Known (49−17, 49, s)-Nets in Base 4

(49−17, 49, 130)-Net over F₄ — Constructive and digital

Digital (32, 49, 130)-net over F₄, using

3 times m-reduction [i] based on digital (32, 52, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 26, 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]

(49−17, 49, 171)-Net over F₄ — Digital

Digital (32, 49, 171)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁹, 171, F₄, 17) (dual of [171, 122, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴⁹, 255, F₄, 17) (dual of [255, 206, 18]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(49−17, 49, 5133)-Net in Base 4 — Upper bound on s

There is no (32, 49, 5134)-net in base 4, because

1 times m-reduction [i] would yield (32, 48, 5134)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 79298 530862 374087 550418 775399 > 4⁴⁸ [i]