MinT - Best Known (83−23, 83, s)-Nets in Base 32

Best Known (83−23, 83, s)-Nets in Base 32

(83−23, 83, 3055)-Net over F₃₂ — Constructive and digital

Digital (60, 83, 3055)-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 (44, 67, 2979)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶⁷, 2979, F₃₂, 23, 23) (dual of [(2979, 23), 68450, 24]-NRT-code), using
    - OOA 11-folding and stacking with additional row [i] based on linear OA(32⁶⁷, 32770, F₃₂, 23) (dual of [32770, 32703, 24]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁶⁷, 32771, F₃₂, 23) (dual of [32771, 32704, 24]-code), using
        constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
        linear OA(32⁶⁷, 32768, F₃₂, 23) (dual of [32768, 32701, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(32⁶⁴, 32768, F₃₂, 22) (dual of [32768, 32704, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(32⁰, 3, F₃₂, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(83−23, 83, 23832)-Net in Base 32 — Constructive

(60, 83, 23832)-net in base 32, using

net defined by OOA [i] based on OOA(32⁸³, 23832, S₃₂, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(32⁸³, 262153, S₃₂, 23), using
  - discarding factors based on OA(32⁸³, 262155, S₃₂, 23), using
    - discarding parts of the base [i] based on linear OA(64⁶⁹, 262155, F₆₄, 23) (dual of [262155, 262086, 24]-code), using
      - constructionÂ X applied to C_e(22) âŠ‚ C_e(19) [i] based on
        linear OA(64⁶⁷, 262144, F₆₄, 23) (dual of [262144, 262077, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
        linear OA(64⁵⁸, 262144, F₆₄, 20) (dual of [262144, 262086, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(64², 11, F₆₄, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,64)), using
        discarding factors / shortening the dual code based on linear OA(64², 64, F₆₄, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
        Reedâ€“Solomon code RS(62,64) [i]

(83−23, 83, 139330)-Net over F₃₂ — Digital

Digital (60, 83, 139330)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−23, 83, large)-Net in Base 32 — Upper bound on s

There is no (60, 83, large)-net in base 32, because

21 times m-reduction [i] would yield (60, 62, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (83−23, 83, s)-Nets in Base 32

(83−23, 83, 3055)-Net over F32 — Constructive and digital

(83−23, 83, 23832)-Net in Base 32 — Constructive

(83−23, 83, 139330)-Net over F32 — Digital

(83−23, 83, large)-Net in Base 32 — Upper bound on s

(83−23, 83, 3055)-Net over F₃₂ — Constructive and digital

(83−23, 83, 139330)-Net over F₃₂ — Digital