MinT - Best Known (30, 131, s)-Nets in Base 5

Best Known (30, 131, s)-Nets in Base 5

(30, 131, 51)-Net over F₅ — Constructive and digital

Digital (30, 131, 51)-net over F₅, using

t-expansion [i] based on digital (22, 131, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(30, 131, 58)-Net over F₅ — Digital

Digital (30, 131, 58)-net over F₅, using

net from sequence [i] based on digital (30, 57)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 30 and N(F) ≥ 58, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(30, 131, 191)-Net over F₅ — Upper bound on s (digital)

There is no digital (30, 131, 192)-net over F₅, because

1 times m-reduction [i] would yield digital (30, 130, 192)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹³⁰, 192, F₅, 100) (dual of [192, 62, 101]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(5¹²⁹, 148, S₅, 100), but
      - the linear programming bound shows that MÂ â‰¥ 437019 493054 884386 993159 269280 163944 474914 010322 994172 449412 913421 078197 284685 984413 954429 328441 619873 046875 / 236583 689197 696731 > 5¹²⁹ [i]
    2. OA(5⁶², 192, S₅, 44), but
      - discarding factors would yield OA(5⁶², 190, S₅, 44), but
        the linear programming bound shows that MÂ â‰¥ 5 531619 630605 621733 791634 558919 444104 118885 526484 067787 791939 510637 894272 804260 253906 250000 000000 000000 / 236416 842360 352611 941840 260955 732881 660879 805306 461094 587839 > 5⁶² [i]

(30, 131, 284)-Net in Base 5 — Upper bound on s

There is no (30, 131, 285)-net in base 5, because

1 times m-reduction [i] would yield (30, 130, 285)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 8 232151 378498 065941 569974 312281 968429 931138 562286 792790 136512 002558 506638 633473 253837 622105 > 5¹³⁰ [i]