MinT - Best Known (64, 90, s)-Nets in Base 32

Best Known (64, 90, s)-Nets in Base 32

(64, 90, 2564)-Net over F₃₂ — Constructive and digital

Digital (64, 90, 2564)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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 (50, 76, 2520)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷⁶, 2520, F₃₂, 26, 26) (dual of [(2520, 26), 65444, 27]-NRT-code), using
    - OA 13-folding and stacking [i] based on linear OA(32⁷⁶, 32760, F₃₂, 26) (dual of [32760, 32684, 27]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁷⁶, 32768, F₃₂, 26) (dual of [32768, 32692, 27]-code), using
        an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]

(64, 90, 5042)-Net in Base 32 — Constructive

(64, 90, 5042)-net in base 32, using

1 times m-reduction [i] based on (64, 91, 5042)-net in base 32, using
- base change [i] based on (38, 65, 5042)-net in base 128, using
  - 128¹ times duplication [i] based on (37, 64, 5042)-net in base 128, using
    - base change [i] based on digital (29, 56, 5042)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁵⁶, 5042, F₂₅₆, 27, 27) (dual of [(5042, 27), 136078, 28]-NRT-code), using
        OOA 13-folding and stacking with additional row [i] based on linear OA(256⁵⁶, 65547, F₂₅₆, 27) (dual of [65547, 65491, 28]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁵⁶, 65548, F₂₅₆, 27) (dual of [65548, 65492, 28]-code), using
        constructionÂ X applied to C([0,13]) âŠ‚ C([0,11]) [i] based on
        linear OA(256⁵³, 65537, F₂₅₆, 27) (dual of [65537, 65484, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(256⁴⁵, 65537, F₂₅₆, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(256³, 11, F₂₅₆, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
        discarding factors / shortening the dual code based on linear OA(256³, 256, F₂₅₆, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
        Reedâ€“Solomon code RS(253,256) [i]

(64, 90, 86073)-Net over F₃₂ — Digital

Digital (64, 90, 86073)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 90, large)-Net in Base 32 — Upper bound on s

There is no (64, 90, large)-net in base 32, because

24 times m-reduction [i] would yield (64, 66, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (64, 90, s)-Nets in Base 32

(64, 90, 2564)-Net over F32 — Constructive and digital

(64, 90, 5042)-Net in Base 32 — Constructive

(64, 90, 86073)-Net over F32 — Digital

(64, 90, large)-Net in Base 32 — Upper bound on s

(64, 90, 2564)-Net over F₃₂ — Constructive and digital

(64, 90, 86073)-Net over F₃₂ — Digital