MinT - Best Known (4, 4+106, s)-Nets in Base 25

Best Known (4, 4+106, s)-Nets in Base 25

(4, 4+106, 66)-Net over F₂₅ — Constructive and digital

Digital (4, 110, 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+106, 127)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (4, 110, 128)-net over F₂₅, because

6 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+106, 146)-Net in Base 25 — Upper bound on s

There is no (4, 110, 147)-net in base 25, because

extracting embedded orthogonal array [i] would yield OA(25¹¹⁰, 147, S₂₅, 106), but
- the linear programming bound shows that MÂ â‰¥ 7652 286119 518652 577096 452328 998918 316918 185962 597643 063592 230006 125978 099455 043462 480441 379956 453523 560432 489127 385670 196410 116794 538680 551340 803503 990173 339843 750000 / 1 288529 486657 > 25¹¹⁰ [i]