MinT - Best Known (54−21, 54, s)-Nets in Base 32

Best Known (54−21, 54, s)-Nets in Base 32

(54−21, 54, 256)-Net over F₃₂ — Constructive and digital

Digital (33, 54, 256)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 7, 64)-net over F₃₂, using
  - s-reduction based on digital (2, 7, 496)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32⁷, 496, F₃₂, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (3, 10, 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]
3. digital (3, 13, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)
4. digital (3, 24, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(54−21, 54, 516)-Net in Base 32 — Constructive

(33, 54, 516)-net in base 32, using

base change [i] based on (24, 45, 516)-net in base 64, using
- 64¹ times duplication [i] based on (23, 44, 516)-net in base 64, using
  - base change [i] based on digital (12, 33, 516)-net over F₂₅₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (1, 11, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
      2. digital (1, 22, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(54−21, 54, 3113)-Net over F₃₂ — Digital

Digital (33, 54, 3113)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−21, 54, large)-Net in Base 32 — Upper bound on s

There is no (33, 54, large)-net in base 32, because

19 times m-reduction [i] would yield (33, 35, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]