MinT - Best Known (90, 90+31, s)-Nets in Base 3

Best Known (90, 90+31, s)-Nets in Base 3

(90, 90+31, 264)-Net over F₃ — Constructive and digital

Digital (90, 121, 264)-net over F₃, using

3¹ times duplication [i] based on digital (89, 120, 264)-net over F₃, using
- trace code for nets [i] based on digital (9, 40, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(90, 90+31, 525)-Net over F₃ — Digital

Digital (90, 121, 525)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²¹, 525, F₃, 31) (dual of [525, 404, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²¹, 728, F₃, 31) (dual of [728, 607, 32]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−6,−5,…,24}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]

(90, 90+31, 21057)-Net in Base 3 — Upper bound on s

There is no (90, 121, 21058)-net in base 3, because

1 times m-reduction [i] would yield (90, 120, 21058)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1797 587211 423437 993691 089376 213409 543283 235030 587568 694681 > 3¹²⁰ [i]