MinT - Best Known (45−28, 45, s)-Nets in Base 256

Best Known (45−28, 45, s)-Nets in Base 256

(45−28, 45, 517)-Net over F₂₅₆ — Constructive and digital

Digital (17, 45, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 15, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 30, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(45−28, 45, 610)-Net over F₂₅₆ — Digital

Digital (17, 45, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁵, 610, F₂₅₆, 3, 28) (dual of [(610, 3), 1785, 29]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁵, 289, F₂₅₆, 3, 14) (dual of [(289, 3), 852, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,852P) [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(256³⁰, 321, F₂₅₆, 3, 28) (dual of [(321, 3), 933, 29]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,934P) [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]

(45−28, 45, 1305228)-Net in Base 256 — Upper bound on s

There is no (17, 45, 1305229)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 348544 057432 387454 172704 174962 714174 580307 374235 363792 891869 768411 902322 788472 721099 528982 815277 111458 414656 > 256⁴⁵ [i]