MinT - Best Known (77−64, 77, s)-Nets in Base 5

Best Known (77−64, 77, s)-Nets in Base 5

(77−64, 77, 34)-Net over F₅ — Constructive and digital

Digital (13, 77, 34)-net over F₅, using

net from sequence [i] based on digital (13, 33)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 11, N(F)Â =Â 32, and 2 places with degree 2 [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(77−64, 77, 36)-Net over F₅ — Digital

Digital (13, 77, 36)-net over F₅, using

net from sequence [i] based on digital (13, 35)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 13 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(77−64, 77, 72)-Net over F₅ — Upper bound on s (digital)

There is no digital (13, 77, 73)-net over F₅, because

9 times m-reduction [i] would yield digital (13, 68, 73)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5⁶⁸, 73, F₅, 55) (dual of [73, 5, 56]-code), but
  - residual code [i] would yield linear OA(5¹³, 17, F₅, 11) (dual of [17, 4, 12]-code), but
    - a result by Boukliev, Kapralov, Maruta, and Fukui [i]

(77−64, 77, 74)-Net in Base 5 — Upper bound on s

There is no (13, 77, 75)-net in base 5, because

11 times m-reduction [i] would yield (13, 66, 75)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁶⁶, 75, S₅, 53), but
  - the linear programming bound shows that MÂ â‰¥ 4684 092198 316280 875047 823428 758420 050144 195556 640625 / 235521 > 5⁶⁶ [i]