MinT - Best Known (78, 78+27, s)-Nets in Base 3

Best Known (78, 78+27, s)-Nets in Base 3

Digital (78, 105, 252)-net over F₃, using

trace code for nets [i] based on digital (8, 35, 84)-net over F₂₇, using
- net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
    - a function field by SÃ©mirat [i]

Digital (78, 105, 459)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰⁵, 459, F₃, 27) (dual of [459, 354, 28]-code), using
- 353 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0) [i] based on linear OA(3²⁷, 28, F₃, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
  - dual of repetition code with length 28 [i]

There is no (78, 105, 18581)-net in base 3, because

1 times m-reduction [i] would yield (78, 104, 18581)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41 764134 282557 466195 133060 173270 445568 509460 666907 > 3¹⁰⁴ [i]