MinT - Best Known (51−22, 51, s)-Nets in Base 25

Best Known (51−22, 51, s)-Nets in Base 25

Digital (29, 51, 208)-net over F₂₅, using

Digital (29, 51, 909)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵¹, 909, F₂₅, 22) (dual of [909, 858, 23]-code), using
- 854 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 9 times 0, 1, 11 times 0, 1, 13 times 0, 1, 15 times 0, 1, 18 times 0, 1, 22 times 0, 1, 25 times 0, 1, 31 times 0, 1, 35 times 0, 1, 42 times 0, 1, 49 times 0, 1, 57 times 0, 1, 67 times 0, 1, 79 times 0, 1, 91 times 0, 1, 108 times 0, 1, 126 times 0) [i] based on linear OA(25²², 26, F₂₅, 22) (dual of [26, 4, 23]-code or 26-arc in PG(21,25)), using
  - extended Reedâ€“Solomon code RS_e(4,25) [i]
  - algebraic-geometric code AG(F, Q+0P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using the rational function field F₂₅(x) [i]
  - algebraic-geometric code AG(F,1P) with degPÂ =Â 3 [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26 (see above)

There is no (29, 51, 619666)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 197217 008487 466723 819747 447437 807414 644011 085618 221647 035161 699031 058385 > 25⁵¹ [i]