MinT - Best Known (73−37, 73, s)-Nets in Base 32

Best Known (73−37, 73, s)-Nets in Base 32

(73−37, 73, 218)-Net over F₃₂ — Constructive and digital

Digital (36, 73, 218)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 25, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
2. digital (11, 48, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(73−37, 73, 288)-Net in Base 32 — Constructive

(36, 73, 288)-net in base 32, using

t-expansion [i] based on (35, 73, 288)-net in base 32, using
- 18 times m-reduction [i] based on (35, 91, 288)-net in base 32, using
  - base change [i] based on digital (9, 65, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(73−37, 73, 544)-Net over F₃₂ — Digital

Digital (36, 73, 544)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷³, 544, F₃₂, 37) (dual of [544, 471, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁷³, 1023, F₃₂, 37) (dual of [1023, 950, 38]-code), using
  - the primitive BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â {3,4,…,39}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]

(73−37, 73, 255477)-Net in Base 32 — Upper bound on s

There is no (36, 73, 255478)-net in base 32, because

1 times m-reduction [i] would yield (36, 72, 255478)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 348607 564193 037141 186157 902056 589573 478024 252053 695832 579848 984392 252305 660342 669949 367830 463281 221428 457254 > 32⁷² [i]