MinT - Best Known (97, 97+33, s)-Nets in Base 3

Best Known (97, 97+33, s)-Nets in Base 3

(97, 97+33, 282)-Net over F₃ — Constructive and digital

Digital (97, 130, 282)-net over F₃, using

3¹ times duplication [i] based on digital (96, 129, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 43, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(97, 97+33, 574)-Net over F₃ — Digital

Digital (97, 130, 574)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁰, 574, F₃, 33) (dual of [574, 444, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁰, 728, F₃, 33) (dual of [728, 598, 34]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {333,334,…,365}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(97, 97+33, 23879)-Net in Base 3 — Upper bound on s

There is no (97, 130, 23880)-net in base 3, because

1 times m-reduction [i] would yield (97, 129, 23880)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35 392975 043496 103271 946584 942967 248018 760391 824065 282429 244417 > 3¹²⁹ [i]