MinT - Best Known (29−12, 29, s)-Nets in Base 64

Best Known (29−12, 29, s)-Nets in Base 64

(29−12, 29, 748)-Net over F₆₄ — Constructive and digital

Digital (17, 29, 748)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 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 (11, 23, 683)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²³, 683, F₆₄, 12, 12) (dual of [(683, 12), 8173, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(64²³, 4098, F₆₄, 12) (dual of [4098, 4075, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        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]
        linear OA(64²¹, 4096, F₆₄, 11) (dual of [4096, 4075, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(29−12, 29, 2731)-Net in Base 64 — Constructive

(17, 29, 2731)-net in base 64, using

64¹ times duplication [i] based on (16, 28, 2731)-net in base 64, using
- base change [i] based on digital (12, 24, 2731)-net over F₁₂₈, using
  - 128¹ times duplication [i] based on digital (11, 23, 2731)-net over F₁₂₈, using
    - net defined by OOA [i] based on linear OOA(128²³, 2731, F₁₂₈, 12, 12) (dual of [(2731, 12), 32749, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(128²³, 16386, F₁₂₈, 12) (dual of [16386, 16363, 13]-code), using
        constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
        linear OA(128²³, 16384, F₁₂₈, 12) (dual of [16384, 16361, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128²¹, 16384, F₁₂₈, 11) (dual of [16384, 16363, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
        linear OA(128⁰, 2, F₁₂₈, 0) (dual of [2, 2, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(128⁰, s, F₁₂₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(29−12, 29, 4996)-Net over F₆₄ — Digital

Digital (17, 29, 4996)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁹, 4996, F₆₄, 12) (dual of [4996, 4967, 13]-code), using
- 892 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 7 times 0, 1, 44 times 0, 1, 187 times 0, 1, 650 times 0) [i] based on linear OA(64²³, 4098, F₆₄, 12) (dual of [4098, 4075, 13]-code), using
  - constructionÂ X applied to C_e(11) âŠ‚ C_e(10) [i] based on
    1. 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]
    2. linear OA(64²¹, 4096, F₆₄, 11) (dual of [4096, 4075, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(29−12, 29, 5463)-Net in Base 64

(17, 29, 5463)-net in base 64, using

64¹ times duplication [i] based on (16, 28, 5463)-net in base 64, using
- base change [i] based on digital (12, 24, 5463)-net over F₁₂₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128²⁴, 5463, F₁₂₈, 3, 12) (dual of [(5463, 3), 16365, 13]-NRT-code), using
    - OOA 3-folding [i] based on linear OA(128²⁴, 16389, F₁₂₈, 12) (dual of [16389, 16365, 13]-code), using
      - constructionÂ X applied to C_e(11) âŠ‚ C_e(9) [i] based on
        linear OA(128²³, 16384, F₁₂₈, 12) (dual of [16384, 16361, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(128¹⁹, 16384, F₁₂₈, 10) (dual of [16384, 16365, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 128²âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
        linear OA(128¹, 5, F₁₂₈, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(128¹, s, F₁₂₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(29−12, 29, large)-Net in Base 64 — Upper bound on s

There is no (17, 29, large)-net in base 64, because

10 times m-reduction [i] would yield (17, 19, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]