MinT - Best Known (4, 4+105, s)-Nets in Base 25

Best Known (4, 4+105, s)-Nets in Base 25

(4, 4+105, 66)-Net over F₂₅ — Constructive and digital

Digital (4, 109, 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]

(4, 4+105, 127)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (4, 109, 128)-net over F₂₅, because

5 times m-reduction [i] would yield digital (4, 104, 128)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25¹⁰⁴, 128, F₂₅, 100) (dual of [128, 24, 101]-code), but
  - residual code [i] would yield linear OA(25⁴, 27, F₂₅, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,25)), but
    - bound for MDS codes [i]

(4, 4+105, 262)-Net in Base 25 — Upper bound on s

There is no (4, 109, 263)-net in base 25, because

23 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]