MinT - Best Known (37−23, 37, s)-Nets in Base 256

Best Known (37−23, 37, s)-Nets in Base 256

(37−23, 37, 517)-Net over F₂₅₆ — Constructive and digital

Digital (14, 37, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 25, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(37−23, 37, 610)-Net over F₂₅₆ — Digital

Digital (14, 37, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256³⁷, 610, F₂₅₆, 3, 23) (dual of [(610, 3), 1793, 24]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹², 289, F₂₅₆, 3, 11) (dual of [(289, 3), 855, 12]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,855P) [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(256²⁵, 321, F₂₅₆, 3, 23) (dual of [(321, 3), 938, 24]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(3;F,939P) [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]

(37−23, 37, 1465497)-Net in Base 256 — Upper bound on s

There is no (14, 37, 1465498)-net in base 256, because

1 times m-reduction [i] would yield (14, 36, 1465498)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 497 324938 112065 657084 349088 576222 401210 901455 329060 845895 244988 160735 944906 755808 267016 > 256³⁶ [i]