MinT - Best Known (99, 136, s)-Nets in Base 4

Best Known (99, 136, s)-Nets in Base 4

(99, 136, 531)-Net over F₄ — Constructive and digital

Digital (99, 136, 531)-net over F₄, using

2 times m-reduction [i] based on digital (99, 138, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 46, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(99, 136, 948)-Net over F₄ — Digital

Digital (99, 136, 948)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁶, 948, F₄, 37) (dual of [948, 812, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³⁶, 1023, F₄, 37) (dual of [1023, 887, 38]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]

(99, 136, 82486)-Net in Base 4 — Upper bound on s

There is no (99, 136, 82487)-net in base 4, because

1 times m-reduction [i] would yield (99, 135, 82487)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1897 381564 307546 774492 043704 659400 936698 147893 454054 046723 254196 542814 943499 670485 > 4¹³⁵ [i]