MinT - Best Known (31, 31+13, s)-Nets in Base 64

Best Known (31, 31+13, s)-Nets in Base 64

(31, 31+13, 43771)-Net over F₆₄ — Constructive and digital

Digital (31, 44, 43771)-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 (24, 37, 43691)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64³⁷, 43691, F₆₄, 13, 13) (dual of [(43691, 13), 567946, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(64³⁷, 262147, F₆₄, 13) (dual of [262147, 262110, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(64³⁷, 262144, F₆₄, 13) (dual of [262144, 262107, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
        linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 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]

(31, 31+13, 349525)-Net in Base 64 — Constructive

(31, 44, 349525)-net in base 64, using

net defined by OOA [i] based on OOA(64⁴⁴, 349525, S₆₄, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(64⁴⁴, 2097151, S₆₄, 13), using
  - discarding factors based on OA(64⁴⁴, 2097155, S₆₄, 13), using
    - discarding parts of the base [i] based on linear OA(128³⁷, 2097155, F₁₂₈, 13) (dual of [2097155, 2097118, 14]-code), using
      - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(128³⁷, 2097152, F₁₂₈, 13) (dual of [2097152, 2097115, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 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]

(31, 31+13, 352118)-Net over F₆₄ — Digital

Digital (31, 44, 352118)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+13, 423033)-Net in Base 64

(31, 44, 423033)-net in base 64, using

net defined by OOA [i] based on OOA(64⁴⁴, 423033, S₆₄, 15, 13), using
- OOA 2-folding and stacking with additional row [i] based on OOA(64⁴⁴, 846067, S₆₄, 3, 13), using
  - discarding factors based on OOA(64⁴⁴, 846068, S₆₄, 3, 13), using
    - discarding parts of the base [i] based on linear OOA(128³⁷, 846068, F₁₂₈, 3, 13) (dual of [(846068, 3), 2538167, 14]-NRT-code), using
      - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(128³⁷, 846068, F₁₂₈, 2, 13) (dual of [(846068, 2), 1692099, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(128³⁷, 1048577, F₁₂₈, 2, 13) (dual of [(1048577, 2), 2097117, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(128³⁷, 2097154, F₁₂₈, 13) (dual of [2097154, 2097117, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(128³⁷, 2097155, F₁₂₈, 13) (dual of [2097155, 2097118, 14]-code), using
        constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
        linear OA(128³⁷, 2097152, F₁₂₈, 13) (dual of [2097152, 2097115, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [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(128⁰, 3, F₁₂₈, 0) (dual of [3, 3, 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]

(31, 31+13, large)-Net in Base 64 — Upper bound on s

There is no (31, 44, large)-net in base 64, because

11 times m-reduction [i] would yield (31, 33, large)-net in base 64, but
- mutually orthogonal hypercube bound [i]