MinT - Best Known (64, 76, s)-Nets in Base 32

Best Known (64, 76, s)-Nets in Base 32

(64, 76, 1410048)-Net over F₃₂ — Constructive and digital

Digital (64, 76, 1410048)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (14, 20, 11948)-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 (10, 16, 10923)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32¹⁶, 10923, F₃₂, 6, 6) (dual of [(10923, 6), 65522, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(32¹⁶, 32769, F₃₂, 6) (dual of [32769, 32753, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹⁶, 32771, F₃₂, 6) (dual of [32771, 32755, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(32¹⁶, 32768, F₃₂, 6) (dual of [32768, 32752, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(32¹³, 32768, F₃₂, 5) (dual of [32768, 32755, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32⁰, 3, F₃₂, 0) (dual of [3, 3, 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 (44, 56, 1398100)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁵⁶, 1398100, F₃₂, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(32⁵⁶, 8388600, F₃₂, 12) (dual of [8388600, 8388544, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁵⁶, large, F₃₂, 12) (dual of [large, large−56, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(64, 76, 1747628)-Net in Base 32 — Constructive

(64, 76, 1747628)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. digital (16, 22, 349528)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²², 349528, F₃₂, 6, 6) (dual of [(349528, 6), 2097146, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(32²², 1048584, F₃₂, 6) (dual of [1048584, 1048562, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²², 1048585, F₃₂, 6) (dual of [1048585, 1048563, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(3) [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(32¹³, 1048576, F₃₂, 4) (dual of [1048576, 1048563, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(32¹, 9, F₃₂, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(32¹, s, F₃₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. (42, 54, 1398100)-net in base 32, using
  - base change [i] based on digital (33, 45, 1398100)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴⁵, 1398100, F₆₄, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
      - OA 6-folding and stacking [i] based on linear OA(64⁴⁵, 8388600, F₆₄, 12) (dual of [8388600, 8388555, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴⁵, large, F₆₄, 12) (dual of [large, large−45, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 64⁴âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(64, 76, large)-Net over F₃₂ — Digital

Digital (64, 76, large)-net over F₃₂, using

5 times m-reduction [i] based on digital (64, 81, large)-net over F₃₂, using
- Gilbertâ€“VarÅ¡amov bound [i]

(64, 76, large)-Net in Base 32 — Upper bound on s

There is no (64, 76, large)-net in base 32, because

10 times m-reduction [i] would yield (64, 66, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (64, 76, s)-Nets in Base 32

(64, 76, 1410048)-Net over F32 — Constructive and digital

(64, 76, 1747628)-Net in Base 32 — Constructive

(64, 76, large)-Net over F32 — Digital

(64, 76, large)-Net in Base 32 — Upper bound on s

(64, 76, 1410048)-Net over F₃₂ — Constructive and digital

(64, 76, large)-Net over F₃₂ — Digital