MinT - Best Known (56−28, 56, s)-Nets in Base 25

Best Known (56−28, 56, s)-Nets in Base 25

(56−28, 56, 200)-Net over F₂₅ — Constructive and digital

Digital (28, 56, 200)-net over F₂₅, using

t-expansion [i] based on digital (25, 56, 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]

(56−28, 56, 387)-Net over F₂₅ — Digital

Digital (28, 56, 387)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25⁵⁶, 387, F₂₅, 28) (dual of [387, 331, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(25⁵⁶, 636, F₂₅, 28) (dual of [636, 580, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(22) [i] based on
    1. linear OA(25⁵², 625, F₂₅, 28) (dual of [625, 573, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(25⁴⁵, 625, F₂₅, 23) (dual of [625, 580, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    3. linear OA(25⁴, 11, F₂₅, 4) (dual of [11, 7, 5]-code or 11-arc in PG(3,25)), using
      - discarding factors / shortening the dual code based on linear OA(25⁴, 25, F₂₅, 4) (dual of [25, 21, 5]-code or 25-arc in PG(3,25)), using
        Reedâ€“Solomon code RS(21,25) [i]

(56−28, 56, 98395)-Net in Base 25 — Upper bound on s

There is no (28, 56, 98396)-net in base 25, because

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]