MinT - Best Known (117, 205, s)-Nets in Base 2

Best Known (117, 205, s)-Nets in Base 2

(117, 205, 62)-Net over F₂ — Constructive and digital

Digital (117, 205, 62)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 63, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 142, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(117, 205, 80)-Net over F₂ — Digital

Digital (117, 205, 80)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(117, 205, 362)-Net over F₂ — Upper bound on s (digital)

There is no digital (117, 205, 363)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁰⁵, 363, F₂, 88) (dual of [363, 158, 89]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(2²⁰⁴, 299, S₂, 88), but
    - adding a parity check bit [i] would yield OA(2²⁰⁵, 300, S₂, 89), but
      - the linear programming bound shows that MÂ â‰¥ 416927 463188 475542 795463 940731 898983 181308 104641 213119 598295 609827 594303 154474 074874 094714 331435 368448 / 5584 401256 756961 297780 513782 298869 514125 > 2²⁰⁵ [i]
  2. linear OA(2¹⁵⁸, 363, F₂, 64) (dual of [363, 205, 65]-code), but
    - the improved Johnson bound shows that NÂ â‰¤ 1214 271568 452769 866939 025066 626311 211727 927188 642255 823141 634304 < 2²⁰⁵ [i]

(117, 205, 374)-Net in Base 2 — Upper bound on s

There is no (117, 205, 375)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 53 430801 671686 355025 890169 121466 558963 919592 764411 474887 620566 > 2²⁰⁵ [i]