MinT - Best Known (56−49, 56, s)-Nets in Base 256

Best Known (56−49, 56, s)-Nets in Base 256

(56−49, 56, 264)-Net over F₂₅₆ — Constructive and digital

Digital (7, 56, 264)-net over F₂₅₆, using

net from sequence [i] based on digital (7, 263)-sequence over F₂₅₆, using
- Niederreiter sequence [i]

(56−49, 56, 321)-Net over F₂₅₆ — Digital

Digital (7, 56, 321)-net over F₂₅₆, using

t-expansion [i] based on digital (2, 56, 321)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [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]

(56−49, 56, 12685)-Net in Base 256 — Upper bound on s

There is no (7, 56, 12686)-net in base 256, because

1 times m-reduction [i] would yield (7, 55, 12686)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 2 841656 739751 727806 015466 361262 937936 795739 914796 875456 620522 699929 666441 089330 869555 938032 173856 415758 014431 105211 362339 776639 641096 > 256⁵⁵ [i]