MinT - Best Known (105−29, 105, s)-Nets in Base 32

Best Known (105−29, 105, s)-Nets in Base 32

(105−29, 105, 2417)-Net over F₃₂ — Constructive and digital

Digital (76, 105, 2417)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (57, 86, 2341)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸⁶, 2341, F₃₂, 29, 29) (dual of [(2341, 29), 67803, 30]-NRT-code), using
    - OOA 14-folding and stacking with additional row [i] based on linear OA(32⁸⁶, 32775, F₃₂, 29) (dual of [32775, 32689, 30]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁸⁶, 32776, F₃₂, 29) (dual of [32776, 32690, 30]-code), using
        constructionÂ X applied to C([0,14]) âŠ‚ C([0,13]) [i] based on
        linear OA(32⁸⁵, 32769, F₃₂, 29) (dual of [32769, 32684, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
        linear OA(32⁷⁹, 32769, F₃₂, 27) (dual of [32769, 32690, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(32¹, 7, F₃₂, 1) (dual of [7, 6, 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]

(105−29, 105, 18725)-Net in Base 32 — Constructive

(76, 105, 18725)-net in base 32, using

32¹ times duplication [i] based on (75, 104, 18725)-net in base 32, using
- net defined by OOA [i] based on OOA(32¹⁰⁴, 18725, S₃₂, 29, 29), using
  - OOA 14-folding and stacking with additional row [i] based on OA(32¹⁰⁴, 262151, S₃₂, 29), using
    - discarding factors based on OA(32¹⁰⁴, 262152, S₃₂, 29), using
      - discarding parts of the base [i] based on linear OA(64⁸⁶, 262152, F₆₄, 29) (dual of [262152, 262066, 30]-code), using
        constructionÂ X applied to C([0,14]) âŠ‚ C([0,13]) [i] based on
        linear OA(64⁸⁵, 262145, F₆₄, 29) (dual of [262145, 262060, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
        linear OA(64⁷⁹, 262145, F₆₄, 27) (dual of [262145, 262066, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
        linear OA(64¹, 7, F₆₄, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(105−29, 105, 160692)-Net over F₃₂ — Digital

Digital (76, 105, 160692)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(105−29, 105, large)-Net in Base 32 — Upper bound on s

There is no (76, 105, large)-net in base 32, because

27 times m-reduction [i] would yield (76, 78, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (105−29, 105, s)-Nets in Base 32

(105−29, 105, 2417)-Net over F32 — Constructive and digital

(105−29, 105, 18725)-Net in Base 32 — Constructive

(105−29, 105, 160692)-Net over F32 — Digital

(105−29, 105, large)-Net in Base 32 — Upper bound on s

(105−29, 105, 2417)-Net over F₃₂ — Constructive and digital

(105−29, 105, 160692)-Net over F₃₂ — Digital