MinT - Best Known (61−9, 61, s)-Nets in Base 32

Best Known (61−9, 61, s)-Nets in Base 32

(61−9, 61, 4195325)-Net over F₃₂ — Constructive and digital

Digital (52, 61, 4195325)-net over F₃₂, using

generalized (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 (12, 16, 2097150)-net over F₃₂, using
  - s-reduction based on digital (12, 16, 4194301)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32¹⁶, 4194301, F₃₂, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(32¹⁶, 8388602, F₃₂, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹⁶, large, F₃₂, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
3. 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]

(61−9, 61, 4198397)-Net in Base 32 — Constructive

(52, 61, 4198397)-net in base 32, using

32¹ times duplication [i] based on (51, 60, 4198397)-net in base 32, using
- base change [i] based on digital (41, 50, 4198397)-net over F₆₄, using
  - generalized (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 4097)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64⁴, 4097, F₆₄, 3, 3) (dual of [(4097, 3), 12287, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(64⁴, 4097, F₆₄, 2, 3) (dual of [(4097, 2), 8190, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(64⁴, 4097, F₆₄, 3) (dual of [4097, 4093, 4]-code or 4097-cap in PG(3,64)), using
        ovoid in PG(3,Â 64) [i]
    2. digital (9, 13, 2097150)-net over F₆₄, using
      - s-reduction based on digital (9, 13, 4194301)-net over F₆₄, using
        net defined by OOA [i] based on linear OOA(64¹³, 4194301, F₆₄, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(64¹³, 4194301, F₆₄, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(64¹³, 8388602, F₆₄, 4) (dual of [8388602, 8388589, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹³, large, F₆₄, 4) (dual of [large, large−13, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    3. digital (24, 33, 2097150)-net over F₆₄, using
      - net defined by OOA [i] based on linear OOA(64³³, 2097150, F₆₄, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
        OOA 4-folding and stacking with additional row [i] based on linear OA(64³³, 8388601, F₆₄, 9) (dual of [8388601, 8388568, 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]

(61−9, 61, large)-Net over F₃₂ — Digital

Digital (52, 61, large)-net over F₃₂, using

5 times m-reduction [i] based on digital (52, 66, large)-net over F₃₂, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁶⁶, large, F₃₂, 14) (dual of [large, large−66, 15]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(61−9, 61, large)-Net in Base 32 — Upper bound on s

There is no (52, 61, large)-net in base 32, because

7 times m-reduction [i] would yield (52, 54, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (61−9, 61, s)-Nets in Base 32

(61−9, 61, 4195325)-Net over F32 — Constructive and digital

(61−9, 61, 4198397)-Net in Base 32 — Constructive

(61−9, 61, large)-Net over F32 — Digital

(61−9, 61, large)-Net in Base 32 — Upper bound on s

(61−9, 61, 4195325)-Net over F₃₂ — Constructive and digital

(61−9, 61, large)-Net over F₃₂ — Digital