MinT - Best Known (51, 89, s)-Nets in Base 64

Best Known (51, 89, s)-Nets in Base 64

(51, 89, 617)-Net over F₆₄ — Constructive and digital

Digital (51, 89, 617)-net over F₆₄, using

2 times m-reduction [i] based on digital (51, 91, 617)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 23, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (28, 68, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(51, 89, 862)-Net in Base 64 — Constructive

(51, 89, 862)-net in base 64, using

1 times m-reduction [i] based on (51, 90, 862)-net in base 64, using
- net defined by OOA [i] based on OOA(64⁹⁰, 862, S₆₄, 39, 39), using
  - OOA 19-folding and stacking with additional row [i] based on OA(64⁹⁰, 16379, S₆₄, 39), using
    - discarding factors based on OA(64⁹⁰, 16386, S₆₄, 39), using
      - discarding parts of the base [i] based on linear OA(128⁷⁷, 16386, F₁₂₈, 39) (dual of [16386, 16309, 40]-code), using
        constructionÂ X applied to C_e(38) âŠ‚ C_e(37) [i] based on
        linear OA(128⁷⁷, 16384, F₁₂₈, 39) (dual of [16384, 16307, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
        linear OA(128⁷⁵, 16384, F₁₂₈, 38) (dual of [16384, 16309, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(51, 89, 5188)-Net over F₆₄ — Digital

Digital (51, 89, 5188)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁸⁹, 5188, F₆₄, 38) (dual of [5188, 5099, 39]-code), using
- 1076 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 1, 4 times 0, 1, 9 times 0, 1, 21 times 0, 1, 43 times 0, 1, 85 times 0, 1, 163 times 0, 1, 291 times 0, 1, 450 times 0) [i] based on linear OA(64⁷⁵, 4098, F₆₄, 38) (dual of [4098, 4023, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(36) [i] based on
    1. linear OA(64⁷⁵, 4096, F₆₄, 38) (dual of [4096, 4021, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(64⁷³, 4096, F₆₄, 37) (dual of [4096, 4023, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 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]

(51, 89, large)-Net in Base 64 — Upper bound on s

There is no (51, 89, large)-net in base 64, because

36 times m-reduction [i] would yield (51, 53, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]