MinT - Best Known (38−7, 38, s)-Nets in Base 32

Best Known (38−7, 38, s)-Nets in Base 32

(38−7, 38, 3846824)-Net over F₃₂ — Constructive and digital

Digital (31, 38, 3846824)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 7, 1050624)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁷, 1050624, F₃₂, 3, 3) (dual of [(1050624, 3), 3151865, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(32⁷, 1050624, F₃₂, 2, 3) (dual of [(1050624, 2), 2101241, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁷, 1050624, F₃₂, 3) (dual of [1050624, 1050617, 4]-code or 1050624-cap in PG(6,32)), using
        product of two caps with tangent hyperplane [i]
2. digital (24, 31, 2796200)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32³¹, 2796200, F₃₂, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OA(32³¹, 8388601, F₃₂, 7) (dual of [8388601, 8388570, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(32³¹, large, F₃₂, 7) (dual of [large, large−31, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(38−7, 38, 5592400)-Net in Base 32 — Constructive

(31, 38, 5592400)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (5, 8, 3214400)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸, 3214400, F₃₂, 3, 3) (dual of [(3214400, 3), 9643192, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(32⁸, 3214400, F₃₂, 2, 3) (dual of [(3214400, 2), 6428792, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(32⁸, 3214400, F₃₂, 3) (dual of [3214400, 3214392, 4]-code or 3214400-cap in PG(7,32)), using
        product of projective cap with cap in AG(4,Â 32) [i] based on linear OA(32⁴, 1025, F₃₂, 3) (dual of [1025, 1021, 4]-code or 1025-cap in PG(3,32)), using
        ovoid in PG(3,Â 32) [i]
2. (23, 30, 2796200)-net in base 32, using
  - base change [i] based on digital (18, 25, 2796200)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²⁵, 2796200, F₆₄, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(64²⁵, 8388601, F₆₄, 7) (dual of [8388601, 8388576, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(64²⁵, large, F₆₄, 7) (dual of [large, large−25, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]

(38−7, 38, large)-Net over F₃₂ — Digital

Digital (31, 38, large)-net over F₃₂, using

32² times duplication [i] based on digital (29, 36, large)-net over F₃₂, using
- t-expansion [i] based on digital (28, 36, large)-net over F₃₂, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32³⁶, large, F₃₂, 8) (dual of [large, large−36, 9]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(38−7, 38, large)-Net in Base 32 — Upper bound on s

There is no (31, 38, large)-net in base 32, because

5 times m-reduction [i] would yield (31, 33, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (38−7, 38, s)-Nets in Base 32

(38−7, 38, 3846824)-Net over F32 — Constructive and digital

(38−7, 38, 5592400)-Net in Base 32 — Constructive

(38−7, 38, large)-Net over F32 — Digital

(38−7, 38, large)-Net in Base 32 — Upper bound on s

(38−7, 38, 3846824)-Net over F₃₂ — Constructive and digital

(38−7, 38, large)-Net over F₃₂ — Digital