MinT - Best Known (33−13, 33, s)-Nets in Base 64

Best Known (33−13, 33, s)-Nets in Base 64

(33−13, 33, 763)-Net over F₆₄ — Constructive and digital

Digital (20, 33, 763)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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 (13, 26, 683)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁶, 683, F₆₄, 13, 13) (dual of [(683, 13), 8853, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64²⁶, 4099, F₆₄, 13) (dual of [4099, 4073, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(64²⁶, 4102, F₆₄, 13) (dual of [4102, 4076, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(64²⁵, 4097, F₆₄, 13) (dual of [4097, 4072, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(64²¹, 4097, F₆₄, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 64⁴âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(64¹, 5, F₆₄, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(33−13, 33, 2732)-Net in Base 64 — Constructive

(20, 33, 2732)-net in base 64, using

net defined by OOA [i] based on OOA(64³³, 2732, S₆₄, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(64³³, 16393, S₆₄, 13), using
  - discarding factors based on OA(64³³, 16396, S₆₄, 13), using
    - discarding parts of the base [i] based on linear OA(128²⁸, 16396, F₁₂₈, 13) (dual of [16396, 16368, 14]-code), using
      - constructionÂ X applied to C([0,6]) âŠ‚ C([0,4]) [i] based on
        linear OA(128²⁵, 16385, F₁₂₈, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(128¹⁷, 16385, F₁₂₈, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 128⁴âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
        linear OA(128³, 11, F₁₂₈, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,128) or 11-cap in PG(2,128)), using
        discarding factors / shortening the dual code based on linear OA(128³, 128, F₁₂₈, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
        Reedâ€“Solomon code RS(125,128) [i]

(33−13, 33, 7787)-Net over F₆₄ — Digital

Digital (20, 33, 7787)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(33−13, 33, large)-Net in Base 64 — Upper bound on s

There is no (20, 33, large)-net in base 64, because

11 times m-reduction [i] would yield (20, 22, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]

Best Known (33−13, 33, s)-Nets in Base 64

(33−13, 33, 763)-Net over F64 — Constructive and digital

(33−13, 33, 2732)-Net in Base 64 — Constructive

(33−13, 33, 7787)-Net over F64 — Digital

(33−13, 33, large)-Net in Base 64 — Upper bound on s

(33−13, 33, 763)-Net over F₆₄ — Constructive and digital

(33−13, 33, 7787)-Net over F₆₄ — Digital