MinT - Best Known (21, 21+37, s)-Nets in Base 256

Best Known (21, 21+37, s)-Nets in Base 256

(21, 21+37, 517)-Net over F₂₅₆ — Constructive and digital

Digital (21, 58, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 39, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(21, 21+37, 610)-Net over F₂₅₆ — Digital

Digital (21, 58, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁵⁸, 610, F₂₅₆, 5, 37) (dual of [(610, 5), 2992, 38]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁹, 289, F₂₅₆, 5, 18) (dual of [(289, 5), 1426, 19]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1426P) [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(256³⁹, 321, F₂₅₆, 5, 37) (dual of [(321, 5), 1566, 38]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(5;F,1567P) [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]

(21, 21+37, 1252218)-Net in Base 256 — Upper bound on s

There is no (21, 58, 1252219)-net in base 256, because

1 times m-reduction [i] would yield (21, 57, 1252219)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 186071 961063 100002 341600 591467 347251 873066 666002 386050 555360 207210 301550 755409 419297 860048 522928 081029 973765 992369 432127 112321 625137 101961 > 256⁵⁷ [i]