MinT - Best Known (88, 88+32, s)-Nets in Base 4

Best Known (88, 88+32, s)-Nets in Base 4

(88, 88+32, 531)-Net over F₄ — Constructive and digital

Digital (88, 120, 531)-net over F₄, using

t-expansion [i] based on digital (87, 120, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 40, 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]

(88, 88+32, 959)-Net over F₄ — Digital

Digital (88, 120, 959)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁰, 959, F₄, 32) (dual of [959, 839, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²⁰, 1023, F₄, 32) (dual of [1023, 903, 33]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]

(88, 88+32, 74266)-Net in Base 4 — Upper bound on s

There is no (88, 120, 74267)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 767075 603550 412251 786921 447992 786963 059628 351397 880382 873842 901390 635479 > 4¹²⁰ [i]