MinT - Best Known (38, 38+9, s)-Nets in Base 32

Best Known (38, 38+9, s)-Nets in Base 32

(38, 38+9, 2097646)-Net over F₃₂ — Constructive and digital

Digital (38, 47, 2097646)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 496)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶, 496, F₃₂, 4, 4) (dual of [(496, 4), 1978, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(32⁶, 992, F₃₂, 4) (dual of [992, 986, 5]-code), using
      - 1 times truncation [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 (32, 41, 2097150)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁴¹, 2097150, F₃₂, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32⁴¹, 8388601, F₃₂, 9) (dual of [8388601, 8388560, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁴¹, large, F₃₂, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(38, 38+9, 2098158)-Net in Base 32 — Constructive

(38, 47, 2098158)-net in base 32, using

net defined by OOA [i] based on OOA(32⁴⁷, 2098158, S₃₂, 10, 9), using
- OOA 2-folding and stacking with additional row [i] based on OOA(32⁴⁷, 4196317, S₃₂, 2, 9), using
  - discarding parts of the base [i] based on linear OOA(64³⁹, 4196317, F₆₄, 2, 9) (dual of [(4196317, 2), 8392595, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(64⁶, 2016, F₆₄, 2, 4) (dual of [(2016, 2), 4026, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
        1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
      2. linear OOA(64³³, 4194301, F₆₄, 2, 9) (dual of [(4194301, 2), 8388569, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64³³, 8388602, F₆₄, 9) (dual of [8388602, 8388569, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(64³³, large, F₆₄, 9) (dual of [large, large−33, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(38, 38+9, large)-Net over F₃₂ — Digital

Digital (38, 47, large)-net over F₃₂, using

32¹ times duplication [i] based on digital (37, 46, large)-net over F₃₂, using
- t-expansion [i] based on digital (36, 46, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁴⁶, large, F₃₂, 10) (dual of [large, large−46, 11]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(38, 38+9, large)-Net in Base 32 — Upper bound on s

There is no (38, 47, large)-net in base 32, because

7 times m-reduction [i] would yield (38, 40, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]