MinT - Best Known (106−28, 106, s)-Nets in Base 4

Best Known (106−28, 106, s)-Nets in Base 4

(106−28, 106, 531)-Net over F₄ — Constructive and digital

Digital (78, 106, 531)-net over F₄, using

4¹ times duplication [i] based on digital (77, 105, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 35, 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]

(106−28, 106, 930)-Net over F₄ — Digital

Digital (78, 106, 930)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁰⁶, 930, F₄, 28) (dual of [930, 824, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁰⁶, 1023, F₄, 28) (dual of [1023, 917, 29]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â {−6,−5,…,21}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(106−28, 106, 72899)-Net in Base 4 — Upper bound on s

There is no (78, 106, 72900)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6582 726689 392338 439697 400540 793876 570652 199097 220778 618421 773616 > 4¹⁰⁶ [i]