MinT - Best Known (132, 238, s)-Nets in Base 2

Best Known (132, 238, s)-Nets in Base 2

(132, 238, 66)-Net over F₂ — Constructive and digital

Digital (132, 238, 66)-net over F₂, using

2 times m-reduction [i] based on digital (132, 240, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 93, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (39, 147, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(132, 238, 84)-Net over F₂ — Digital

Digital (132, 238, 84)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(132, 238, 334)-Net over F₂ — Upper bound on s (digital)

There is no digital (132, 238, 335)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²³⁸, 335, F₂, 106) (dual of [335, 97, 107]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(2²³⁷, 299, S₂, 106), but
    - the linear programming bound shows that MÂ â‰¥ 2 372623 864838 720824 144751 616230 912568 605364 469958 081012 522169 374458 839809 772727 298051 538944 / 10 001354 293176 388857 > 2²³⁷ [i]
  2. OA(2⁹⁷, 335, S₂, 36), but
    - discarding factors would yield OA(2⁹⁷, 324, S₂, 36), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 158757 007614 184387 760434 595956 > 2⁹⁷ [i]

(132, 238, 389)-Net in Base 2 — Upper bound on s

There is no (132, 238, 390)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 459926 634048 295129 196262 794727 023247 399016 223217 871697 382212 648168 333864 > 2²³⁸ [i]