MinT - Best Known (91−20, 91, s)-Nets in Base 32

Best Known (91−20, 91, s)-Nets in Base 32

(91−20, 91, 104922)-Net over F₃₂ — Constructive and digital

Digital (71, 91, 104922)-net over F₃₂, using

32¹ times duplication [i] based on digital (70, 90, 104922)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 13, 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 (57, 77, 104858)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁷⁷, 104858, F₃₂, 20, 20) (dual of [(104858, 20), 2097083, 21]-NRT-code), using
      - OA 10-folding and stacking [i] based on linear OA(32⁷⁷, 1048580, F₃₂, 20) (dual of [1048580, 1048503, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
        linear OA(32⁷⁷, 1048576, F₃₂, 20) (dual of [1048576, 1048499, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
        linear OA(32⁷³, 1048576, F₃₂, 19) (dual of [1048576, 1048503, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 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]

(91−20, 91, 209718)-Net in Base 32 — Constructive

(71, 91, 209718)-net in base 32, using

base change [i] based on digital (45, 65, 209718)-net over F₁₂₈, using
- net defined by OOA [i] based on linear OOA(128⁶⁵, 209718, F₁₂₈, 20, 20) (dual of [(209718, 20), 4194295, 21]-NRT-code), using
  - OA 10-folding and stacking [i] based on linear OA(128⁶⁵, 2097180, F₁₂₈, 20) (dual of [2097180, 2097115, 21]-code), using
    - discarding factors / shortening the dual code based on linear OA(128⁶⁵, 2097183, F₁₂₈, 20) (dual of [2097183, 2097118, 21]-code), using
      - constructionÂ X applied to C_e(19) âŠ‚ C_e(11) [i] based on
        linear OA(128⁵⁸, 2097152, F₁₂₈, 20) (dual of [2097152, 2097094, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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(128⁷, 31, F₁₂₈, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁷, 128, F₁₂₈, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
        Reedâ€“Solomon code RS(121,128) [i]

(91−20, 91, 4137433)-Net over F₃₂ — Digital

Digital (71, 91, 4137433)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(91−20, 91, large)-Net in Base 32 — Upper bound on s

There is no (71, 91, large)-net in base 32, because

18 times m-reduction [i] would yield (71, 73, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (91−20, 91, s)-Nets in Base 32

(91−20, 91, 104922)-Net over F32 — Constructive and digital

(91−20, 91, 209718)-Net in Base 32 — Constructive

(91−20, 91, 4137433)-Net over F32 — Digital

(91−20, 91, large)-Net in Base 32 — Upper bound on s

(91−20, 91, 104922)-Net over F₃₂ — Constructive and digital

(91−20, 91, 4137433)-Net over F₃₂ — Digital