MinT - Best Known (131−35, 131, s)-Nets in Base 4

Best Known (131−35, 131, s)-Nets in Base 4

(131−35, 131, 531)-Net over F₄ — Constructive and digital

Digital (96, 131, 531)-net over F₄, using

t-expansion [i] based on digital (95, 131, 531)-net over F₄, using
- 1 times m-reduction [i] based on digital (95, 132, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 44, 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]

(131−35, 131, 1008)-Net over F₄ — Digital

Digital (96, 131, 1008)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³¹, 1008, F₄, 35) (dual of [1008, 877, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³¹, 1023, F₄, 35) (dual of [1023, 892, 36]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(131−35, 131, 96105)-Net in Base 4 — Upper bound on s

There is no (96, 131, 96106)-net in base 4, because

1 times m-reduction [i] would yield (96, 130, 96106)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 852682 972080 992969 177346 292371 980410 462471 360192 542736 890552 612193 016337 360138 > 4¹³⁰ [i]