MinT - Best Known (4, 4+5, s)-Nets in Base 32

Best Known (4, 4+5, s)-Nets in Base 32

(4, 4+5, 529)-Net over F₃₂ — Constructive and digital

Digital (4, 9, 529)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
2. 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]

(4, 4+5, 604)-Net over F₃₂ — Digital

Digital (4, 9, 604)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁹, 604, F₃₂, 5) (dual of [604, 595, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(32⁹, 1023, F₃₂, 5) (dual of [1023, 1014, 6]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(4, 4+5, 2016)-Net in Base 32 — Constructive

(4, 9, 2016)-net in base 32, using

net defined by OOA [i] based on OOA(32⁹, 2016, S₃₂, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(32⁹, 4033, S₃₂, 5), using
  - discarding parts of the base [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
    - a code by Danev and Olsson [i]

(4, 4+5, 47835)-Net in Base 32 — Upper bound on s

There is no (4, 9, 47836)-net in base 32, because

1 times m-reduction [i] would yield (4, 8, 47836)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 099545 882559 > 32⁸ [i]