MinT - Best Known (80−28, 80, s)-Nets in Base 64

Best Known (80−28, 80, s)-Nets in Base 64

(80−28, 80, 690)-Net over F₆₄ — Constructive and digital

Digital (52, 80, 690)-net over F₆₄, using

6 times m-reduction [i] based on digital (52, 86, 690)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 24, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]
  2. digital (28, 62, 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]

(80−28, 80, 4682)-Net in Base 64 — Constructive

(52, 80, 4682)-net in base 64, using

base change [i] based on digital (32, 60, 4682)-net over F₂₅₆, using
- 1 times m-reduction [i] based on digital (32, 61, 4682)-net over F₂₅₆, using
  - net defined by OOA [i] based on linear OOA(256⁶¹, 4682, F₂₅₆, 29, 29) (dual of [(4682, 29), 135717, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(256⁶¹, 65549, F₂₅₆, 29) (dual of [65549, 65488, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(256⁶¹, 65550, F₂₅₆, 29) (dual of [65550, 65489, 30]-code), using
        constructionÂ X applied to C_e(28) âŠ‚ C_e(23) [i] based on
        linear OA(256⁵⁷, 65536, F₂₅₆, 29) (dual of [65536, 65479, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(256⁴⁷, 65536, F₂₅₆, 24) (dual of [65536, 65489, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
        linear OA(256⁴, 14, F₂₅₆, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
        discarding factors / shortening the dual code based on linear OA(256⁴, 256, F₂₅₆, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
        Reedâ€“Solomon code RS(252,256) [i]

(80−28, 80, 38982)-Net over F₆₄ — Digital

Digital (52, 80, 38982)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−28, 80, large)-Net in Base 64 — Upper bound on s

There is no (52, 80, large)-net in base 64, because

26 times m-reduction [i] would yield (52, 54, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (80−28, 80, s)-Nets in Base 64

(80−28, 80, 690)-Net over F64 — Constructive and digital

(80−28, 80, 4682)-Net in Base 64 — Constructive

(80−28, 80, 38982)-Net over F64 — Digital

(80−28, 80, large)-Net in Base 64 — Upper bound on s

(80−28, 80, 690)-Net over F₆₄ — Constructive and digital

(80−28, 80, 38982)-Net over F₆₄ — Digital