MinT - Best Known (128−34, 128, s)-Nets in Base 5

Best Known (128−34, 128, s)-Nets in Base 5

Digital (94, 128, 408)-net over F₅, using

trace code for nets [i] based on digital (30, 64, 204)-net over F₂₅, using
- net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
    - a function field by GarcÃa and Quoos [i]

Digital (94, 128, 1709)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁸, 1709, F₅, 34) (dual of [1709, 1581, 35]-code), using
- 1580 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 41 times 0, 1, 43 times 0, 1, 45 times 0, 1, 47 times 0, 1, 50 times 0, 1, 53 times 0, 1, 55 times 0, 1, 58 times 0, 1, 61 times 0, 1, 65 times 0, 1, 67 times 0, 1, 71 times 0, 1, 75 times 0, 1, 79 times 0) [i] based on linear OA(5³⁴, 35, F₅, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,5)), using
  - dual of repetition code with length 35 [i]

There is no (94, 128, 328627)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 293887 867294 380106 388371 586717 063736 251301 321761 913208 300950 551752 966510 598723 393355 754125 > 5¹²⁸ [i]