MinT - Best Known (73, 86, s)-Nets in Base 32

Best Known (73, 86, s)-Nets in Base 32

(73, 86, 1748651)-Net over F₃₂ — Constructive and digital

Digital (73, 86, 1748651)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 25, 350551)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 1025)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32⁴, 1025, F₃₂, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(32⁴, 1025, F₃₂, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [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. digital (15, 21, 349526)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32²¹, 349526, F₃₂, 6, 6) (dual of [(349526, 6), 2097135, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(32²¹, 1048578, F₃₂, 6) (dual of [1048578, 1048557, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(32²¹, 1048580, F₃₂, 6) (dual of [1048580, 1048559, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(32²¹, 1048576, F₃₂, 6) (dual of [1048576, 1048555, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(32¹⁷, 1048576, F₃₂, 5) (dual of [1048576, 1048559, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32⁰, 4, F₃₂, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. digital (48, 61, 1398100)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶¹, 1398100, F₃₂, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
    - OOA 6-folding and stacking with additional row [i] based on linear OA(32⁶¹, 8388601, F₃₂, 13) (dual of [8388601, 8388540, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁶¹, large, F₃₂, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(73, 86, 2796200)-Net in Base 32 — Constructive

(73, 86, 2796200)-net in base 32, using

32² times duplication [i] based on (71, 84, 2796200)-net in base 32, using
- base change [i] based on digital (57, 70, 2796200)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁷⁰, 2796200, F₆₄, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
    - OOA 3-folding and stacking with additional row [i] based on linear OOA(64⁷⁰, 8388601, F₆₄, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(64⁷⁰, 8388602, F₆₄, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
        (u,Â u+v)-construction [i] based on
        linear OOA(64²¹, 4194301, F₆₄, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64²¹, 8388602, F₆₄, 6) (dual of [8388602, 8388581, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(64²¹, large, F₆₄, 6) (dual of [large, large−21, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OOA(64⁴⁹, 4194301, F₆₄, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(64⁴⁹, 8388602, F₆₄, 13) (dual of [8388602, 8388553, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁹, large, F₆₄, 13) (dual of [large, large−49, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(73, 86, large)-Net over F₃₂ — Digital

Digital (73, 86, large)-net over F₃₂, using

t-expansion [i] based on digital (72, 86, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (72, 91, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(73, 86, large)-Net in Base 32 — Upper bound on s

There is no (73, 86, large)-net in base 32, because

11 times m-reduction [i] would yield (73, 75, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (73, 86, s)-Nets in Base 32

(73, 86, 1748651)-Net over F32 — Constructive and digital

(73, 86, 2796200)-Net in Base 32 — Constructive

(73, 86, large)-Net over F32 — Digital

(73, 86, large)-Net in Base 32 — Upper bound on s

(73, 86, 1748651)-Net over F₃₂ — Constructive and digital

(73, 86, large)-Net over F₃₂ — Digital