MinT - Best Known (16, 16+27, s)-Nets in Base 256

Best Known (16, 16+27, s)-Nets in Base 256

(16, 16+27, 517)-Net over F₂₅₆ — Constructive and digital

Digital (16, 43, 517)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (2, 29, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(16, 16+27, 610)-Net over F₂₅₆ — Digital

Digital (16, 43, 610)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁴³, 610, F₂₅₆, 4, 27) (dual of [(610, 4), 2397, 28]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256¹⁴, 289, F₂₅₆, 4, 13) (dual of [(289, 4), 1142, 14]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1142P) [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, 321, F₂₅₆, 4, 27) (dual of [(321, 4), 1255, 28]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(4;F,1256P) [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]

(16, 16+27, 1340700)-Net in Base 256 — Upper bound on s

There is no (16, 43, 1340701)-net in base 256, because

1 times m-reduction [i] would yield (16, 42, 1340701)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 139985 368058 121415 798379 236215 798694 175637 046900 070256 817896 220518 775412 392037 415927 163264 726793 904016 > 256⁴² [i]