MinT - Best Known (34−16, 34, s)-Nets in Base 64

Best Known (34−16, 34, s)-Nets in Base 64

(34−16, 34, 513)-Net over F₆₄ — Constructive and digital

Digital (18, 34, 513)-net over F₆₄, using

64¹ times duplication [i] based on digital (17, 33, 513)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64³³, 513, F₆₄, 16, 16) (dual of [(513, 16), 8175, 17]-NRT-code), using
  - OA 8-folding and stacking [i] based on linear OA(64³³, 4104, F₆₄, 16) (dual of [4104, 4071, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(12) [i] based on
      1. linear OA(64³¹, 4096, F₆₄, 16) (dual of [4096, 4065, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      2. linear OA(64²⁵, 4096, F₆₄, 13) (dual of [4096, 4071, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      3. linear OA(64², 8, F₆₄, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(34−16, 34, 515)-Net in Base 64 — Constructive

(18, 34, 515)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (3, 11, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
    - base change [i] based on digital (0, 9, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]
2. (7, 23, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
    - base change [i] based on digital (1, 18, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(34−16, 34, 2053)-Net over F₆₄ — Digital

Digital (18, 34, 2053)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64³⁴, 2053, F₆₄, 2, 16) (dual of [(2053, 2), 4072, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(64³⁴, 4106, F₆₄, 16) (dual of [4106, 4072, 17]-code), using
  - discarding factors / shortening the dual code based on linear OA(64³⁴, 4107, F₆₄, 16) (dual of [4107, 4073, 17]-code), using
    - constructionÂ X applied to C_e(15) âŠ‚ C_e(11) [i] based on
      1. linear OA(64³¹, 4096, F₆₄, 16) (dual of [4096, 4065, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
      2. linear OA(64²³, 4096, F₆₄, 12) (dual of [4096, 4073, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      3. linear OA(64³, 11, F₆₄, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
        discarding factors / shortening the dual code based on linear OA(64³, 64, F₆₄, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
        Reedâ€“Solomon code RS(61,64) [i]

(34−16, 34, 2835396)-Net in Base 64 — Upper bound on s

There is no (18, 34, 2835397)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 25 711020 856547 877009 162981 964585 329978 535043 774405 427321 903448 > 64³⁴ [i]