MinT - Best Known (35−21, 35, s)-Nets in Base 256

Best Known (35−21, 35, s)-Nets in Base 256

(35−21, 35, 518)-Net over F₂₅₆ — Constructive and digital

Digital (14, 35, 518)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 23, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(35−21, 35, 642)-Net over F₂₅₆ — Digital

Digital (14, 35, 642)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁵, 642, F₂₅₆, 2, 21) (dual of [(642, 2), 1249, 22]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹², 321, F₂₅₆, 2, 10) (dual of [(321, 2), 630, 11]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,631P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]
  2. linear OOA(256²³, 321, F₂₅₆, 2, 21) (dual of [(321, 2), 619, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(2;F,620P) [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321 (see above)

(35−21, 35, 2738112)-Net in Base 256 — Upper bound on s

There is no (14, 35, 2738113)-net in base 256, because

1 times m-reduction [i] would yield (14, 34, 2738113)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 7588 564732 059635 387190 695151 565117 336038 472369 065552 124007 504630 943943 041858 671776 > 256³⁴ [i]