MinT - Best Known (39−25, 39, s)-Nets in Base 256

Best Known (39−25, 39, s)-Nets in Base 256

(39−25, 39, 516)-Net over F₂₅₆ — Constructive and digital

Digital (14, 39, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 26, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(39−25, 39, 578)-Net over F₂₅₆ — Digital

Digital (14, 39, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁹, 578, F₂₅₆, 5, 25) (dual of [(578, 5), 2851, 26]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹³, 289, F₂₅₆, 5, 12) (dual of [(289, 5), 1432, 13]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1432P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
      - sharp bound on the number of rational points on curves with genus 1 [i]
  2. linear OOA(256²⁶, 289, F₂₅₆, 5, 25) (dual of [(289, 5), 1419, 26]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1419P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(39−25, 39, 876822)-Net in Base 256 — Upper bound on s

There is no (14, 39, 876823)-net in base 256, because

1 times m-reduction [i] would yield (14, 38, 876823)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 32 592777 385244 382774 474632 792825 583044 746832 923421 810651 970504 506260 191573 599753 004843 384406 > 256³⁸ [i]