MinT - Best Known (6, 6+16, s)-Nets in Base 4

Best Known (6, 6+16, s)-Nets in Base 4

(6, 6+16, 17)-Net over F₄ — Constructive and digital

Digital (6, 22, 17)-net over F₄, using

t-expansion [i] based on digital (5, 22, 17)-net over F₄, using
- net from sequence [i] based on digital (5, 16)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(6, 6+16, 20)-Net over F₄ — Digital

Digital (6, 22, 20)-net over F₄, using

net from sequence [i] based on digital (6, 19)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 6 and N(F) ≥ 20, using
  - Drinfeld modules of rank 1 [i]

(6, 6+16, 46)-Net over F₄ — Upper bound on s (digital)

There is no digital (6, 22, 47)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²², 47, F₄, 16) (dual of [47, 25, 17]-code), but
- residual code [i] would yield OA(4⁶, 30, S₄, 4), but
  - the linear programming bound shows that MÂ â‰¥ 179200 / 43 > 4⁶ [i]

(6, 6+16, 50)-Net in Base 4 — Upper bound on s

There is no (6, 22, 51)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 17 913715 338106 > 4²² [i]