MinT - Best Known (61, 61+23, s)-Nets in Base 64

Best Known (61, 61+23, s)-Nets in Base 64

(61, 61+23, 23976)-Net over F₆₄ — Constructive and digital

Digital (61, 84, 23976)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (6, 17, 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, 12, 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 (44, 67, 23831)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁶⁷, 23831, F₆₄, 23, 23) (dual of [(23831, 23), 548046, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(64⁶⁷, 262142, F₆₄, 23) (dual of [262142, 262075, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁶⁷, 262144, F₆₄, 23) (dual of [262144, 262077, 24]-code), using
        an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(61, 61+23, 190652)-Net in Base 64 — Constructive

(61, 84, 190652)-net in base 64, using

base change [i] based on digital (49, 72, 190652)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128⁷², 190652, F₁₂₈, 23, 23) (dual of [(190652, 23), 4384924, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(128⁷², 2097173, F₁₂₈, 23) (dual of [2097173, 2097101, 24]-code), using
    - discarding factors / shortening the dual code based on linear OA(128⁷², 2097176, F₁₂₈, 23) (dual of [2097176, 2097104, 24]-code), using
      - constructionÂ X applied to C([0,11]) âŠ‚ C([0,8]) [i] based on
        linear OA(128⁶⁷, 2097153, F₁₂₈, 23) (dual of [2097153, 2097086, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(128⁴⁹, 2097153, F₁₂₈, 17) (dual of [2097153, 2097104, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153Â | 128⁶âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
        linear OA(128⁵, 23, F₁₂₈, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁵, 128, F₁₂₈, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
        Reedâ€“Solomon code RS(123,128) [i]

(61, 61+23, 1131984)-Net over F₆₄ — Digital

Digital (61, 84, 1131984)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+23, large)-Net in Base 64 — Upper bound on s

There is no (61, 84, large)-net in base 64, because

21 times m-reduction [i] would yield (61, 63, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (61, 61+23, s)-Nets in Base 64

(61, 61+23, 23976)-Net over F64 — Constructive and digital

(61, 61+23, 190652)-Net in Base 64 — Constructive

(61, 61+23, 1131984)-Net over F64 — Digital

(61, 61+23, large)-Net in Base 64 — Upper bound on s

(61, 61+23, 23976)-Net over F₆₄ — Constructive and digital

(61, 61+23, 1131984)-Net over F₆₄ — Digital