MinT - Best Known (32−15, 32, s)-Nets in Base 7

Best Known (32−15, 32, s)-Nets in Base 7

(32−15, 32, 102)-Net over F₇ — Constructive and digital

Digital (17, 32, 102)-net over F₇, using

trace code for nets [i] based on digital (1, 16, 51)-net over F₄₉, using
- net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(32−15, 32, 128)-Net over F₇ — Digital

Digital (17, 32, 128)-net over F₇, using

trace code for nets [i] based on digital (1, 16, 64)-net over F₄₉, using
- net from sequence [i] based on digital (1, 63)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(32−15, 32, 3110)-Net in Base 7 — Upper bound on s

There is no (17, 32, 3111)-net in base 7, because

1 times m-reduction [i] would yield (17, 31, 3111)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 158 123100 384359 320643 852311 > 7³¹ [i]