MinT - Best Known (21, 21+41, s)-Nets in Base 256

Best Known (21, 21+41, s)-Nets in Base 256

(21, 21+41, 515)-Net over F₂₅₆ — Constructive and digital

Digital (21, 62, 515)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 20, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 42, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(21, 21+41, 546)-Net over F₂₅₆ — Digital

Digital (21, 62, 546)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 20, 257)-net over F₂₅₆, using
  - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
      - the rational function field F₂₅₆(x) [i]
    - Niederreiter sequence [i]
2. digital (1, 42, 289)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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]

(21, 21+41, 720928)-Net in Base 256 — Upper bound on s

There is no (21, 62, 720929)-net in base 256, because

1 times m-reduction [i] would yield (21, 61, 720929)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 799 179451 012722 146190 996019 240578 637144 671004 112051 220227 776430 530749 342494 052455 802168 157825 039720 790540 823554 468849 041292 811599 224468 548709 544776 > 256⁶¹ [i]