MinT - Best Known (46−13, 46, s)-Nets in Base 64

Best Known (46−13, 46, s)-Nets in Base 64

(46−13, 46, 43821)-Net over F₆₄ — Constructive and digital

Digital (33, 46, 43821)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 9, 130)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 6, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (24, 37, 43691)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 43691, F₆₄, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64³⁷, 262147, F₆₄, 13) (dual of [262147, 262110, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(64³⁷, 262144, F₆₄, 13) (dual of [262144, 262107, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(46−13, 46, 349527)-Net in Base 64 — Constructive

(33, 46, 349527)-net in base 64, using

net defined by OOA [i] based on OOA(64⁴⁶, 349527, S₆₄, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(64⁴⁶, 2097163, S₆₄, 13), using
  - discarding parts of the base [i] based on linear OA(128³⁹, 2097163, F₁₂₈, 13) (dual of [2097163, 2097124, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
      1. linear OA(128³⁷, 2097152, F₁₂₈, 13) (dual of [2097152, 2097115, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(128²⁸, 2097152, F₁₂₈, 10) (dual of [2097152, 2097124, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(128², 11, F₁₂₈, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
        discarding factors / shortening the dual code based on linear OA(128², 128, F₁₂₈, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
        Reedâ€“Solomon code RS(126,128) [i]

(46−13, 46, 704230)-Net over F₆₄ — Digital

Digital (33, 46, 704230)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(46−13, 46, large)-Net in Base 64 — Upper bound on s

There is no (33, 46, large)-net in base 64, because

11 times m-reduction [i] would yield (33, 35, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (46−13, 46, s)-Nets in Base 64

(46−13, 46, 43821)-Net over F64 — Constructive and digital

(46−13, 46, 349527)-Net in Base 64 — Constructive

(46−13, 46, 704230)-Net over F64 — Digital

(46−13, 46, large)-Net in Base 64 — Upper bound on s

(46−13, 46, 43821)-Net over F₆₄ — Constructive and digital

(46−13, 46, 704230)-Net over F₆₄ — Digital