MinT - Best Known (28, 57, s)-Nets in Base 25

Best Known (28, 57, s)-Nets in Base 25

(28, 57, 200)-Net over F₂₅ — Constructive and digital

Digital (28, 57, 200)-net over F₂₅, using

t-expansion [i] based on digital (25, 57, 200)-net over F₂₅, using
- net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
    - a function field by GarcÃa and Quoos [i]

(28, 57, 349)-Net over F₂₅ — Digital

Digital (28, 57, 349)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵⁷, 349, F₂₅, 29) (dual of [349, 292, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵⁷, 624, F₂₅, 29) (dual of [624, 567, 30]-code), using
  - the primitive BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â {3,4,…,31}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(28, 57, 98395)-Net in Base 25 — Upper bound on s

There is no (28, 57, 98396)-net in base 25, because

1 times m-reduction [i] would yield (28, 56, 98396)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 1 926064 905545 720184 541570 185454 081334 145671 167004 670410 422154 787948 389811 288385 > 25⁵⁶ [i]