MinT - Best Known (7, 7+10, s)-Nets in Base 64

Best Known (7, 7+10, s)-Nets in Base 64

(7, 7+10, 177)-Net over F₆₄ — Constructive and digital

Digital (7, 17, 177)-net over F₆₄, using

net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
  - a function field by GarcÃa and Quoos [i]

(7, 7+10, 202)-Net over F₆₄ — Digital

Digital (7, 17, 202)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64¹⁷, 202, F₆₄, 10) (dual of [202, 185, 11]-code), using
- 6 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 5 times 0) [i] based on linear OA(64¹⁶, 195, F₆₄, 10) (dual of [195, 179, 11]-code), using
  - the BCH-code C(I) with length 195Â | 64²âˆ’1, defining interval IÂ =Â {1,8,15,…,64}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(7, 7+10, 259)-Net in Base 64 — Constructive

(7, 17, 259)-net in base 64, using

3 times m-reduction [i] based on (7, 20, 259)-net in base 64, using
- base change [i] based on digital (2, 15, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(7, 7+10, 321)-Net in Base 64

(7, 17, 321)-net in base 64, using

3 times m-reduction [i] based on (7, 20, 321)-net in base 64, using
- base change [i] based on digital (2, 15, 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]

(7, 7+10, 57212)-Net in Base 64 — Upper bound on s

There is no (7, 17, 57213)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 5 070771 304122 500904 786672 325240 > 64¹⁷ [i]