MinT - Best Known (27, 27+39, s)-Nets in Base 256

Best Known (27, 27+39, s)-Nets in Base 256

(27, 27+39, 522)-Net over F₂₅₆ — Constructive and digital

Digital (27, 66, 522)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 43, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆ (see above)

(27, 27+39, 1027)-Net over F₂₅₆ — Digital

Digital (27, 66, 1027)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶⁶, 1027, F₂₅₆, 2, 39) (dual of [(1027, 2), 1988, 40]-NRT-code), using
- constructionÂ X applied to AG(2;F,2008P) âŠ‚ AG(2;F,2012P) [i] based on
  1. linear OOA(256⁶³, 1024, F₂₅₆, 2, 39) (dual of [(1024, 2), 1985, 40]-NRT-code), using algebraic-geometric NRT-code AG(2;F,2008P) [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]
  2. linear OOA(256⁵⁹, 1024, F₂₅₆, 2, 35) (dual of [(1024, 2), 1989, 36]-NRT-code), using algebraic-geometric NRT-code AG(2;F,2012P) [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025 (see above)
  3. linear OOA(256³, 3, F₂₅₆, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(256³, 256, F₂₅₆, 2, 3) (dual of [(256, 2), 509, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;509,256) [i]

(27, 27+39, 5387536)-Net in Base 256 — Upper bound on s

There is no (27, 66, 5387537)-net in base 256, because

1 times m-reduction [i] would yield (27, 65, 5387537)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 432398 910977 995301 586719 095926 877623 941355 809654 210158 185090 447063 399513 505156 677271 403592 509098 466828 293694 366853 852580 309328 145230 314712 350011 930592 297216 > 256⁶⁵ [i]