MinT - Best Known (77−21, 77, s)-Nets in Base 64

Best Known (77−21, 77, s)-Nets in Base 64

(77−21, 77, 26359)-Net over F₆₄ — Constructive and digital

Digital (56, 77, 26359)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (6, 16, 145)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 5, 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 (1, 11, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (40, 61, 26214)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁶¹, 26214, F₆₄, 21, 21) (dual of [(26214, 21), 550433, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(64⁶¹, 262141, F₆₄, 21) (dual of [262141, 262080, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁶¹, 262144, F₆₄, 21) (dual of [262144, 262083, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(77−21, 77, 209717)-Net in Base 64 — Constructive

(56, 77, 209717)-net in base 64, using

base change [i] based on digital (45, 66, 209717)-net over F₁₂₈, using
- 128¹ times duplication [i] based on digital (44, 65, 209717)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁶⁵, 209717, F₁₂₈, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁶⁵, 2097171, F₁₂₈, 21) (dual of [2097171, 2097106, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
        linear OA(128⁶¹, 2097152, F₁₂₈, 21) (dual of [2097152, 2097091, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(128⁴⁶, 2097152, F₁₂₈, 16) (dual of [2097152, 2097106, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(128⁴, 19, F₁₂₈, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(77−21, 77, 1185120)-Net over F₆₄ — Digital

Digital (56, 77, 1185120)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(77−21, 77, large)-Net in Base 64 — Upper bound on s

There is no (56, 77, large)-net in base 64, because

19 times m-reduction [i] would yield (56, 58, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (77−21, 77, s)-Nets in Base 64

(77−21, 77, 26359)-Net over F64 — Constructive and digital

(77−21, 77, 209717)-Net in Base 64 — Constructive

(77−21, 77, 1185120)-Net over F64 — Digital

(77−21, 77, large)-Net in Base 64 — Upper bound on s

(77−21, 77, 26359)-Net over F₆₄ — Constructive and digital

(77−21, 77, 1185120)-Net over F₆₄ — Digital