MinT - Best Known (58, 58+49, s)-Nets in Base 16

Best Known (58, 58+49, s)-Nets in Base 16

(58, 58+49, 522)-Net over F₁₆ — Constructive and digital

Digital (58, 107, 522)-net over F₁₆, using

1 times m-reduction [i] based on digital (58, 108, 522)-net over F₁₆, using
- trace code for nets [i] based on digital (4, 54, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(58, 58+49, 644)-Net over F₁₆ — Digital

Digital (58, 107, 644)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(16¹⁰⁷, 644, F₁₆, 2, 49) (dual of [(644, 2), 1181, 50]-NRT-code), using
- 16¹ times duplication [i] based on linear OOA(16¹⁰⁶, 644, F₁₆, 2, 49) (dual of [(644, 2), 1182, 50]-NRT-code), using
  - 2 times NRT-code embedding in larger space [i] based on linear OOA(16¹⁰², 642, F₁₆, 2, 49) (dual of [(642, 2), 1182, 50]-NRT-code), using
    - extracting embedded OOA [i] based on digital (53, 102, 642)-net over F₁₆, using
      - trace code for nets [i] based on digital (2, 51, 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]

(58, 58+49, 135964)-Net in Base 16 — Upper bound on s

There is no (58, 107, 135965)-net in base 16, because

1 times m-reduction [i] would yield (58, 106, 135965)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 43 325119 363694 460645 011824 438290 617407 736005 669244 863533 799537 983951 259548 672095 998991 528669 927804 698517 985208 536357 595562 920651 > 16¹⁰⁶ [i]