MinT - Best Known (44−28, 44, s)-Nets in Base 256

Best Known (44−28, 44, s)-Nets in Base 256

(44−28, 44, 516)-Net over F₂₅₆ — Constructive and digital

Digital (16, 44, 516)-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 (1, 29, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(44−28, 44, 578)-Net over F₂₅₆ — Digital

Digital (16, 44, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴⁴, 578, F₂₅₆, 5, 28) (dual of [(578, 5), 2846, 29]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁵, 289, F₂₅₆, 5, 14) (dual of [(289, 5), 1430, 15]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1430P) [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(256²⁹, 289, F₂₅₆, 5, 28) (dual of [(289, 5), 1416, 29]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1416P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(44−28, 44, 878351)-Net in Base 256 — Upper bound on s

There is no (16, 44, 878352)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 9174 023987 052186 943367 304007 285817 445840 353531 308753 536314 083964 074386 135823 054356 166739 266104 795370 828591 > 256⁴⁴ [i]