MinT - Best Known (46, 46+44, s)-Nets in Base 27

Best Known (46, 46+44, s)-Nets in Base 27

(46, 46+44, 192)-Net over F₂₇ — Constructive and digital

Digital (46, 90, 192)-net over F₂₇, using

4 times m-reduction [i] based on digital (46, 94, 192)-net over F₂₇, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 35, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
        an explicitly constructive algebraic function field [i]
  2. digital (11, 59, 96)-net over F₂₇, using
    - net from sequence [i] based on digital (11, 95)-sequence over F₂₇ (see above)

(46, 46+44, 370)-Net in Base 27 — Constructive

(46, 90, 370)-net in base 27, using

t-expansion [i] based on (43, 90, 370)-net in base 27, using
- 18 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(46, 46+44, 666)-Net over F₂₇ — Digital

Digital (46, 90, 666)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁹⁰, 666, F₂₇, 44) (dual of [666, 576, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁹⁰, 749, F₂₇, 44) (dual of [749, 659, 45]-code), using
  - constructionÂ X applied to C_e(43) âŠ‚ C_e(36) [i] based on
    1. linear OA(27⁸⁴, 729, F₂₇, 44) (dual of [729, 645, 45]-code), using an extension C_e(43) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,43], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 44 [i]
    2. linear OA(27⁷⁰, 729, F₂₇, 37) (dual of [729, 659, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(27⁶, 20, F₂₇, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,27)), using
      - discarding factors / shortening the dual code based on linear OA(27⁶, 27, F₂₇, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,27)), using
        Reedâ€“Solomon code RS(21,27) [i]

(46, 46+44, 249711)-Net in Base 27 — Upper bound on s

There is no (46, 90, 249712)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 664 887643 094500 856951 464517 399221 370854 225224 877383 838310 540861 966854 716677 651959 086924 398139 616731 879213 708250 209339 684795 836577 > 27⁹⁰ [i]