MinT - Best Known (92−24, 92, s)-Nets in Base 32

Best Known (92−24, 92, s)-Nets in Base 32

(92−24, 92, 2838)-Net over F₃₂ — Constructive and digital

Digital (68, 92, 2838)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (10, 22, 108)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 7, 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 (3, 15, 64)-net over F₃₂, using
      - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
2. digital (46, 70, 2730)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷⁰, 2730, F₃₂, 24, 24) (dual of [(2730, 24), 65450, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(32⁷⁰, 32760, F₃₂, 24) (dual of [32760, 32690, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷⁰, 32768, F₃₂, 24) (dual of [32768, 32698, 25]-code), using
        an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]

(92−24, 92, 21847)-Net in Base 32 — Constructive

(68, 92, 21847)-net in base 32, using

32² times duplication [i] based on (66, 90, 21847)-net in base 32, using
- base change [i] based on digital (51, 75, 21847)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷⁵, 21847, F₆₄, 24, 24) (dual of [(21847, 24), 524253, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(64⁷⁵, 262164, F₆₄, 24) (dual of [262164, 262089, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁷⁵, 262167, F₆₄, 24) (dual of [262167, 262092, 25]-code), using
        constructionÂ X applied to C_e(23) âŠ‚ C_e(17) [i] based on
        linear OA(64⁷⁰, 262144, F₆₄, 24) (dual of [262144, 262074, 25]-code), using an extension C_e(23) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,23], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 24 [i]
        linear OA(64⁵², 262144, F₆₄, 18) (dual of [262144, 262092, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
        linear OA(64⁵, 23, F₆₄, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,64)), using
        discarding factors / shortening the dual code based on linear OA(64⁵, 64, F₆₄, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
        Reedâ€“Solomon code RS(59,64) [i]

(92−24, 92, 318941)-Net over F₃₂ — Digital

Digital (68, 92, 318941)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(92−24, 92, large)-Net in Base 32 — Upper bound on s

There is no (68, 92, large)-net in base 32, because

22 times m-reduction [i] would yield (68, 70, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (92−24, 92, s)-Nets in Base 32

(92−24, 92, 2838)-Net over F32 — Constructive and digital

(92−24, 92, 21847)-Net in Base 32 — Constructive

(92−24, 92, 318941)-Net over F32 — Digital

(92−24, 92, large)-Net in Base 32 — Upper bound on s

(92−24, 92, 2838)-Net over F₃₂ — Constructive and digital

(92−24, 92, 318941)-Net over F₃₂ — Digital