MinT - Best Known (22, 31, s)-Nets in Base 32

Best Known (22, 31, s)-Nets in Base 32

(22, 31, 8688)-Net over F₃₂ — Constructive and digital

Digital (22, 31, 8688)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 6, 496)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁶, 496, F₃₂, 4, 4) (dual of [(496, 4), 1978, 5]-NRT-code), using
    - OA 2-folding and stacking [i] based on linear OA(32⁶, 992, F₃₂, 4) (dual of [992, 986, 5]-code), using
      - 1 times truncation [i] based on linear OA(32⁷, 993, F₃₂, 5) (dual of [993, 986, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (16, 25, 8192)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁵, 8192, F₃₂, 9, 9) (dual of [(8192, 9), 73703, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(32²⁵, 32769, F₃₂, 9) (dual of [32769, 32744, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(22, 31, 65536)-Net in Base 32 — Constructive

(22, 31, 65536)-net in base 32, using

32¹ times duplication [i] based on (21, 30, 65536)-net in base 32, using
- base change [i] based on digital (16, 25, 65536)-net over F₆₄, using
  - net defined by OOA [i] based on linear OOA(64²⁵, 65536, F₆₄, 9, 9) (dual of [(65536, 9), 589799, 10]-NRT-code), using
    - OOA 4-folding and stacking with additional row [i] based on linear OA(64²⁵, 262145, F₆₄, 9) (dual of [262145, 262120, 10]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(22, 31, 82567)-Net over F₃₂ — Digital

Digital (22, 31, 82567)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(22, 31, 131073)-Net in Base 32

(22, 31, 131073)-net in base 32, using

32¹ times duplication [i] based on (21, 30, 131073)-net in base 32, using
- base change [i] based on digital (16, 25, 131073)-net over F₆₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64²⁵, 131073, F₆₄, 2, 9) (dual of [(131073, 2), 262121, 10]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(64²⁵, 262146, F₆₄, 9) (dual of [262146, 262121, 10]-code), using
      - discarding factors / shortening the dual code based on linear OA(64²⁵, 262147, F₆₄, 9) (dual of [262147, 262122, 10]-code), using
        constructionÂ X applied to C_e(8) âŠ‚ C_e(7) [i] based on
        linear OA(64²⁵, 262144, F₆₄, 9) (dual of [262144, 262119, 10]-code), using an extension C_e(8) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
        linear OA(64²², 262144, F₆₄, 8) (dual of [262144, 262122, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(22, 31, large)-Net in Base 32 — Upper bound on s

There is no (22, 31, large)-net in base 32, because

7 times m-reduction [i] would yield (22, 24, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]