MinT - Best Known (99−95, 99, s)-Nets in Base 25

Best Known (99−95, 99, s)-Nets in Base 25

(99−95, 99, 66)-Net over F₂₅ — Constructive and digital

Digital (4, 99, 66)-net over F₂₅, using

net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
  - a function field by GarcÃa and Quoos [i]

(99−95, 99, 163)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (4, 99, 164)-net over F₂₅, because

5 times m-reduction [i] would yield digital (4, 94, 164)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25⁹⁴, 164, F₂₅, 90) (dual of [164, 70, 91]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(25⁹³, 97, S₂₅, 90), but
      - the linear programming bound shows that MÂ â‰¥ 917 366046 106511 319204 310488 250559 166131 581647 062117 601772 786429 813888 045207 279411 397468 873458 869997 872255 908077 931962 907314 300537 109375 / 81263 > 25⁹³ [i]
    2. linear OA(25⁷⁰, 164, F₂₅, 67) (dual of [164, 94, 68]-code), but
      - discarding factors / shortening the dual code would yield linear OA(25⁷⁰, 148, F₂₅, 67) (dual of [148, 78, 68]-code), but
        constructionÂ Y1 [i] would yield
        OA(25⁶⁹, 73, S₂₅, 67), but
        the linear programming bound shows that MÂ â‰¥ 24178 564222 846123 668546 399721 917528 895562 151649 914153 732205 158997 548011 257094 913162 291049 957275 390625 / 7242 > 25⁶⁹ [i]
        linear OA(25⁷⁸, 148, F₂₅, 75) (dual of [148, 70, 76]-code), but
        discarding factors / shortening the dual code would yield linear OA(25⁷⁸, 103, F₂₅, 75) (dual of [103, 25, 76]-code), but
        residual code [i] would yield OA(25³, 27, S₂₅, 3), but
        bound for OAs with index unity [i]

(99−95, 99, 262)-Net in Base 25 — Upper bound on s

There is no (4, 99, 263)-net in base 25, because

13 times m-reduction [i] would yield (4, 86, 263)-net in base 25, but
- extracting embedded orthogonal array [i] would yield OA(25⁸⁶, 263, S₂₅, 82), but
  - the linear programming bound shows that MÂ â‰¥ 124838 157887 948575 069973 603564 217894 140049 811879 259470 500094 671199 192064 814369 803279 087877 393316 825789 390511 262347 445431 419856 731736 217625 439167 022705 078125 / 73787 952480 415645 668068 103685 901541 > 25⁸⁶ [i]