MinT - Best Known (48−14, 48, s)-Nets in Base 32

Best Known (48−14, 48, s)-Nets in Base 32

(48−14, 48, 4725)-Net over F₃₂ — Constructive and digital

Digital (34, 48, 4725)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (26, 40, 4681)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴⁰, 4681, F₃₂, 14, 14) (dual of [(4681, 14), 65494, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(32⁴⁰, 32767, F₃₂, 14) (dual of [32767, 32727, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴⁰, 32768, F₃₂, 14) (dual of [32768, 32728, 15]-code), using
        an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(48−14, 48, 37449)-Net in Base 32 — Constructive

(34, 48, 37449)-net in base 32, using

base change [i] based on digital (26, 40, 37449)-net over F₆₄, using
- net defined by OOA [i] based on linear OOA(64⁴⁰, 37449, F₆₄, 14, 14) (dual of [(37449, 14), 524246, 15]-NRT-code), using
  - OA 7-folding and stacking [i] based on linear OA(64⁴⁰, 262143, F₆₄, 14) (dual of [262143, 262103, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴⁰, 262144, F₆₄, 14) (dual of [262144, 262104, 15]-code), using
      - an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(48−14, 48, 66003)-Net over F₃₂ — Digital

Digital (34, 48, 66003)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−14, 48, 131073)-Net in Base 32

(34, 48, 131073)-net in base 32, using

base change [i] based on digital (26, 40, 131073)-net over F₆₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64⁴⁰, 131073, F₆₄, 2, 14) (dual of [(131073, 2), 262106, 15]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(64⁴⁰, 262146, F₆₄, 14) (dual of [262146, 262106, 15]-code), using
    - discarding factors / shortening the dual code based on linear OA(64⁴⁰, 262147, F₆₄, 14) (dual of [262147, 262107, 15]-code), using
      - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
        linear OA(64⁴⁰, 262144, F₆₄, 14) (dual of [262144, 262104, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
        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(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]

(48−14, 48, large)-Net in Base 32 — Upper bound on s

There is no (34, 48, large)-net in base 32, because

12 times m-reduction [i] would yield (34, 36, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]