MinT - Best Known (167−29, 167, s)-Nets in Base 3

Best Known (167−29, 167, s)-Nets in Base 3

(167−29, 167, 707)-Net over F₃ — Constructive and digital

Digital (138, 167, 707)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 23, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (115, 144, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 36, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(167−29, 167, 4660)-Net over F₃ — Digital

Digital (138, 167, 4660)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁷, 4660, F₃, 29) (dual of [4660, 4493, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁷, 6576, F₃, 29) (dual of [6576, 6409, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
      - dual of repetition code with length 15 [i]
    2. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(167−29, 167, 1373151)-Net in Base 3 — Upper bound on s

There is no (138, 167, 1373152)-net in base 3, because

1 times m-reduction [i] would yield (138, 166, 1373152)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 926925 765241 940489 791714 867929 554880 057798 933629 150648 658054 604794 172129 570113 > 3¹⁶⁶ [i]