MinT - Best Known (43, 86, s)-Nets in Base 32

Best Known (43, 86, s)-Nets in Base 32

(43, 86, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 86, 240)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 32, 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]
2. digital (11, 54, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(43, 86, 513)-Net in Base 32 — Constructive

(43, 86, 513)-net in base 32, using

4 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(43, 86, 669)-Net over F₃₂ — Digital

Digital (43, 86, 669)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁸⁶, 669, F₃₂, 43) (dual of [669, 583, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁸⁶, 1030, F₃₂, 43) (dual of [1030, 944, 44]-code), using
  - constructionÂ X applied to C([0,21]) âŠ‚ C([0,20]) [i] based on
    1. linear OA(32⁸⁵, 1025, F₃₂, 43) (dual of [1025, 940, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,21], and minimum distance dÂ â‰¥ |{−21,−20,…,21}|+1Â =Â 44 (BCH-bound) [i]
    2. linear OA(32⁸¹, 1025, F₃₂, 41) (dual of [1025, 944, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025Â | 32⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
    3. linear OA(32¹, 5, F₃₂, 1) (dual of [5, 4, 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]

(43, 86, 346251)-Net in Base 32 — Upper bound on s

There is no (43, 86, 346252)-net in base 32, because

1 times m-reduction [i] would yield (43, 85, 346252)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 86 650484 647066 069421 397684 957516 530886 024130 287933 669977 060311 416047 903941 562882 596395 546280 107569 672626 676459 024217 320937 389808 > 32⁸⁵ [i]