MinT - Best Known (76−20, 76, s)-Nets in Base 64

Best Known (76−20, 76, s)-Nets in Base 64

(76−20, 76, 26409)-Net over F₆₄ — Constructive and digital

Digital (56, 76, 26409)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 18, 195)-net over F₆₄, using
  - 1 times m-reduction [i] based on digital (8, 19, 195)-net over F₆₄, using
    - generalized (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, 5, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
      3. digital (0, 11, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)
2. digital (38, 58, 26214)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁵⁸, 26214, F₆₄, 20, 20) (dual of [(26214, 20), 524222, 21]-NRT-code), using
    - OA 10-folding and stacking [i] based on linear OA(64⁵⁸, 262140, F₆₄, 20) (dual of [262140, 262082, 21]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁵⁸, 262144, F₆₄, 20) (dual of [262144, 262086, 21]-code), using
        an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(76−20, 76, 209718)-Net in Base 64 — Constructive

(56, 76, 209718)-net in base 64, using

net defined by OOA [i] based on OOA(64⁷⁶, 209718, S₆₄, 20, 20), using
- OA 10-folding and stacking [i] based on OA(64⁷⁶, 2097180, S₆₄, 20), using
  - discarding factors based on OA(64⁷⁶, 2097183, S₆₄, 20), using
    - discarding parts of the base [i] based on linear OA(128⁶⁵, 2097183, F₁₂₈, 20) (dual of [2097183, 2097118, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(11) [i] based on
        linear OA(128⁵⁸, 2097152, F₁₂₈, 20) (dual of [2097152, 2097094, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(128³⁴, 2097152, F₁₂₈, 12) (dual of [2097152, 2097118, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128⁷, 31, F₁₂₈, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁷, 128, F₁₂₈, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
        Reedâ€“Solomon code RS(121,128) [i]

(76−20, 76, 2111528)-Net over F₆₄ — Digital

Digital (56, 76, 2111528)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(76−20, 76, large)-Net in Base 64 — Upper bound on s

There is no (56, 76, large)-net in base 64, because

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

Best Known (76−20, 76, s)-Nets in Base 64

(76−20, 76, 26409)-Net over F64 — Constructive and digital

(76−20, 76, 209718)-Net in Base 64 — Constructive

(76−20, 76, 2111528)-Net over F64 — Digital

(76−20, 76, large)-Net in Base 64 — Upper bound on s

(76−20, 76, 26409)-Net over F₆₄ — Constructive and digital

(76−20, 76, 2111528)-Net over F₆₄ — Digital