MinT - Best Known (65−42, 65, s)-Nets in Base 256

Best Known (65−42, 65, s)-Nets in Base 256

(65−42, 65, 516)-Net over F₂₅₆ — Constructive and digital

Digital (23, 65, 516)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (1, 43, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(65−42, 65, 578)-Net over F₂₅₆ — Digital

Digital (23, 65, 578)-net over F₂₅₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(256⁶⁵, 578, F₂₅₆, 6, 42) (dual of [(578, 6), 3403, 43]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(256²², 289, F₂₅₆, 6, 21) (dual of [(289, 6), 1712, 22]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1712P) [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(256⁴³, 289, F₂₅₆, 6, 42) (dual of [(289, 6), 1691, 43]-NRT-code), using
    - extended algebraic-geometric NRT-code AG_e(6;F,1691P) [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289 (see above)

(65−42, 65, 968331)-Net in Base 256 — Upper bound on s

There is no (23, 65, 968332)-net in base 256, because

the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 432454 595904 680950 894862 454948 359997 005102 161847 719671 966019 633090 517858 408964 244583 502418 971698 434028 986932 694715 309124 154625 774084 644309 908181 702281 426736 > 256⁶⁵ [i]