MinT - Best Known (89, 89+13, s)-Nets in Base 32

Best Known (89, 89+13, s)-Nets in Base 32

(89, 89+13, 3320523)-Net over F₃₂ — Constructive and digital

Digital (89, 102, 3320523)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (11, 15, 524323)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 2, 33)-net over F₃₂, using
      - kÂ =Â 2 construction [i]
    2. digital (9, 13, 524290)-net over F₃₂, using
      - net defined by OOA [i] based on linear OOA(32¹³, 524290, F₃₂, 4, 4) (dual of [(524290, 4), 2097147, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(32¹³, 1048580, F₃₂, 4) (dual of [1048580, 1048567, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        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(32⁹, 1048576, F₃₂, 3) (dual of [1048576, 1048567, 4]-code or 1048576-cap in PG(8,32)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 1048575Â = 32⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [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 (20, 26, 1398100)-net over F₃₂, using
  - s-reduction based on digital (20, 26, 2796201)-net over F₃₂, using
    - net defined by OOA [i] based on linear OOA(32²⁶, 2796201, F₃₂, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(32²⁶, large, F₃₂, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 33554431Â = 32⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
3. 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]

(89, 89+13, 4194301)-Net in Base 32 — Constructive

(89, 102, 4194301)-net in base 32, using

base change [i] based on digital (72, 85, 4194301)-net over F₆₄, using
- 64² times duplication [i] based on digital (70, 83, 4194301)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64⁸³, 4194301, F₆₄, 15, 13) (dual of [(4194301, 15), 62914432, 14]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(64⁸³, large, F₆₄, 3, 13), using
      - generalized (u,Â u+v)-construction [i] based on
        linear OOA(64¹³, 2796201, F₆₄, 3, 4) (dual of [(2796201, 3), 8388590, 5]-NRT-code), using
        discarding factors / shortening the dual code 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]
        linear OOA(64²¹, 2796201, F₆₄, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
        OOA 3-folding [i] 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, 2796201, F₆₄, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
        OOA 3-folding [i] 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]

(89, 89+13, large)-Net over F₃₂ — Digital

Digital (89, 102, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 102, large)-net over F₃₂, using
- 8 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+13, large)-Net in Base 32 — Upper bound on s

There is no (89, 102, large)-net in base 32, because

11 times m-reduction [i] would yield (89, 91, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 89+13, s)-Nets in Base 32

(89, 89+13, 3320523)-Net over F32 — Constructive and digital

(89, 89+13, 4194301)-Net in Base 32 — Constructive

(89, 89+13, large)-Net over F32 — Digital

(89, 89+13, large)-Net in Base 32 — Upper bound on s

(89, 89+13, 3320523)-Net over F₃₂ — Constructive and digital

(89, 89+13, large)-Net over F₃₂ — Digital