MinT - Best Known (89−20, 89, s)-Nets in Base 32

Best Known (89−20, 89, s)-Nets in Base 32

(89−20, 89, 104902)-Net over F₃₂ — Constructive and digital

Digital (69, 89, 104902)-net over F₃₂, using

32¹ times duplication [i] based on digital (68, 88, 104902)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 11, 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 (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]

(89−20, 89, 209717)-Net in Base 32 — Constructive

(69, 89, 209717)-net in base 32, using

32² times duplication [i] based on (67, 87, 209717)-net in base 32, using
- net defined by OOA [i] based on OOA(32⁸⁷, 209717, S₃₂, 20, 20), using
  - OA 10-folding and stacking [i] based on OA(32⁸⁷, 2097170, S₃₂, 20), using
    - discarding factors based on OA(32⁸⁷, 2097171, S₃₂, 20), using
      - discarding parts of the base [i] based on linear OA(128⁶², 2097171, F₁₂₈, 20) (dual of [2097171, 2097109, 21]-code), using
        constructionÂ X applied to C_e(19) âŠ‚ C_e(14) [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(128⁴³, 2097152, F₁₂₈, 15) (dual of [2097152, 2097109, 16]-code), using an extension C_e(14) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]
        linear OA(128⁴, 19, F₁₂₈, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(89−20, 89, 2872729)-Net over F₃₂ — Digital

Digital (69, 89, 2872729)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−20, 89, large)-Net in Base 32 — Upper bound on s

There is no (69, 89, large)-net in base 32, because

18 times m-reduction [i] would yield (69, 71, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (89−20, 89, s)-Nets in Base 32

(89−20, 89, 104902)-Net over F32 — Constructive and digital

(89−20, 89, 209717)-Net in Base 32 — Constructive

(89−20, 89, 2872729)-Net over F32 — Digital

(89−20, 89, large)-Net in Base 32 — Upper bound on s

(89−20, 89, 104902)-Net over F₃₂ — Constructive and digital

(89−20, 89, 2872729)-Net over F₃₂ — Digital