MinT - Best Known (13, 13+9, s)-Nets in Base 4

Best Known (13, 13+9, s)-Nets in Base 4

(13, 13+9, 66)-Net over F₄ — Constructive and digital

Digital (13, 22, 66)-net over F₄, using

trace code for nets [i] based on digital (2, 11, 33)-net over F₁₆, using
- net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
    - a function field by SÃ©mirat [i]

(13, 13+9, 68)-Net over F₄ — Digital

Digital (13, 22, 68)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²², 68, F₄, 9) (dual of [68, 46, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4²², 85, F₄, 9) (dual of [85, 63, 10]-code), using
  - the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {−20,−17,−14,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]

(13, 13+9, 1065)-Net in Base 4 — Upper bound on s

There is no (13, 22, 1066)-net in base 4, because

1 times m-reduction [i] would yield (13, 21, 1066)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4 404622 917046 > 4²¹ [i]