MinT - Best Known (82, 82+23, s)-Nets in Base 32

Best Known (82, 82+23, s)-Nets in Base 32

(82, 82+23, 95401)-Net over F₃₂ — Constructive and digital

Digital (82, 105, 95401)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 16, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (66, 89, 95325)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸⁹, 95325, F₃₂, 23, 23) (dual of [(95325, 23), 2192386, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(32⁸⁹, 1048576, F₃₂, 23) (dual of [1048576, 1048487, 24]-code), using
      - an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(82, 82+23, 190683)-Net in Base 32 — Constructive

(82, 105, 190683)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. (71, 94, 190650)-net in base 32, using
  - net defined by OOA [i] based on OOA(32⁹⁴, 190650, S₃₂, 23, 23), using
    - OOA 11-folding and stacking with additional row [i] based on OA(32⁹⁴, 2097151, S₃₂, 23), using
      - discarding factors based on OA(32⁹⁴, 2097155, S₃₂, 23), using
        discarding parts of the base [i] based on linear OA(128⁶⁷, 2097155, F₁₂₈, 23) (dual of [2097155, 2097088, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
        linear OA(128⁶⁷, 2097152, F₁₂₈, 23) (dual of [2097152, 2097085, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(128⁶⁴, 2097152, F₁₂₈, 22) (dual of [2097152, 2097088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [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]

(82, 82+23, 4458233)-Net over F₃₂ — Digital

Digital (82, 105, 4458233)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+23, large)-Net in Base 32 — Upper bound on s

There is no (82, 105, large)-net in base 32, because

21 times m-reduction [i] would yield (82, 84, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (82, 82+23, s)-Nets in Base 32

(82, 82+23, 95401)-Net over F32 — Constructive and digital

(82, 82+23, 190683)-Net in Base 32 — Constructive

(82, 82+23, 4458233)-Net over F32 — Digital

(82, 82+23, large)-Net in Base 32 — Upper bound on s

(82, 82+23, 95401)-Net over F₃₂ — Constructive and digital

(82, 82+23, 4458233)-Net over F₃₂ — Digital