MinT - Best Known (45−13, 45, s)-Nets in Base 32

Best Known (45−13, 45, s)-Nets in Base 32

(45−13, 45, 5506)-Net over F₃₂ — Constructive and digital

Digital (32, 45, 5506)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 7, 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 (25, 38, 5462)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32³⁸, 5462, F₃₂, 13, 13) (dual of [(5462, 13), 70968, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(32³⁸, 32773, F₃₂, 13) (dual of [32773, 32735, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(32³⁸, 32776, F₃₂, 13) (dual of [32776, 32738, 14]-code), using
        constructionÂ X applied to C([0,6]) âŠ‚ C([0,5]) [i] based on
        linear OA(32³⁷, 32769, F₃₂, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(32³¹, 32769, F₃₂, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]
        linear OA(32¹, 7, F₃₂, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(45−13, 45, 43691)-Net in Base 32 — Constructive

(32, 45, 43691)-net in base 32, using

net defined by OOA [i] based on OOA(32⁴⁵, 43691, S₃₂, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(32⁴⁵, 262147, S₃₂, 13), using
  - discarding parts of the base [i] based on linear OA(64³⁷, 262147, F₆₄, 13) (dual of [262147, 262110, 14]-code), using
    - constructionÂ X applied to C_e(12) âŠ‚ C_e(11) [i] based on
      1. linear OA(64³⁷, 262144, F₆₄, 13) (dual of [262144, 262107, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
      2. linear OA(64³⁴, 262144, F₆₄, 12) (dual of [262144, 262110, 13]-code), using an extension C_e(11) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]
      3. linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(45−13, 45, 75222)-Net over F₃₂ — Digital

Digital (32, 45, 75222)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45−13, 45, large)-Net in Base 32 — Upper bound on s

There is no (32, 45, large)-net in base 32, because

11 times m-reduction [i] would yield (32, 34, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (45−13, 45, s)-Nets in Base 32

(45−13, 45, 5506)-Net over F32 — Constructive and digital

(45−13, 45, 43691)-Net in Base 32 — Constructive

(45−13, 45, 75222)-Net over F32 — Digital

(45−13, 45, large)-Net in Base 32 — Upper bound on s

(45−13, 45, 5506)-Net over F₃₂ — Constructive and digital

(45−13, 45, 75222)-Net over F₃₂ — Digital