MinT - Best Known (139−35, 139, s)-Nets in Base 3

Best Known (139−35, 139, s)-Nets in Base 3

(139−35, 139, 288)-Net over F₃ — Constructive and digital

Digital (104, 139, 288)-net over F₃, using

3¹ times duplication [i] based on digital (103, 138, 288)-net over F₃, using
- trace code for nets [i] based on digital (11, 46, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [i]

(139−35, 139, 623)-Net over F₃ — Digital

Digital (104, 139, 623)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁹, 623, F₃, 35) (dual of [623, 484, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁹, 728, F₃, 35) (dual of [728, 589, 36]-code), using
  - the primitive BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â {−7,−6,…,27}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(139−35, 139, 26776)-Net in Base 3 — Upper bound on s

There is no (104, 139, 26777)-net in base 3, because

1 times m-reduction [i] would yield (104, 138, 26777)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 696531 667044 975433 351873 876045 724129 343603 680793 981612 830794 047987 > 3¹³⁸ [i]