MinT - Best Known (49, 70, s)-Nets in Base 32

Best Known (49, 70, s)-Nets in Base 32

(49, 70, 3280)-Net over F₃₂ — Constructive and digital

Digital (49, 70, 3280)-net over F₃₂, using

32¹ times duplication [i] based on digital (48, 69, 3280)-net over F₃₂, using
- net defined by OOA [i] based on linear OOA(32⁶⁹, 3280, F₃₂, 21, 21) (dual of [(3280, 21), 68811, 22]-NRT-code), using
  - OOA 10-folding and stacking with additional row [i] based on linear OA(32⁶⁹, 32801, F₃₂, 21) (dual of [32801, 32732, 22]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(32⁶⁸, 32800, F₃₂, 21) (dual of [32800, 32732, 22]-code), using
      - constructionÂ X applied to C([0,10]) âŠ‚ C([0,6]) [i] based on
        linear OA(32⁶¹, 32769, F₃₂, 21) (dual of [32769, 32708, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(32³⁷, 32769, F₃₂, 13) (dual of [32769, 32732, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769Â | 32⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
        linear OA(32⁷, 31, F₃₂, 7) (dual of [31, 24, 8]-code or 31-arc in PG(6,32)), using
        discarding factors / shortening the dual code based on linear OA(32⁷, 32, F₃₂, 7) (dual of [32, 25, 8]-code or 32-arc in PG(6,32)), using
        Reedâ€“Solomon code RS(25,32) [i]

(49, 70, 6554)-Net in Base 32 — Constructive

(49, 70, 6554)-net in base 32, using

base change [i] based on (29, 50, 6554)-net in base 128, using
- 128² times duplication [i] based on (27, 48, 6554)-net in base 128, using
  - base change [i] based on digital (21, 42, 6554)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴², 6554, F₂₅₆, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
      - OOA 10-folding and stacking with additional row [i] based on linear OA(256⁴², 65541, F₂₅₆, 21) (dual of [65541, 65499, 22]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴², 65542, F₂₅₆, 21) (dual of [65542, 65500, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(256⁴¹, 65537, F₂₅₆, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(256³⁷, 65537, F₂₅₆, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(256¹, 5, F₂₅₆, 1) (dual of [5, 4, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(256¹, s, F₂₅₆, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(49, 70, 49666)-Net over F₃₂ — Digital

Digital (49, 70, 49666)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 70, large)-Net in Base 32 — Upper bound on s

There is no (49, 70, large)-net in base 32, because

19 times m-reduction [i] would yield (49, 51, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (49, 70, s)-Nets in Base 32

(49, 70, 3280)-Net over F32 — Constructive and digital

(49, 70, 6554)-Net in Base 32 — Constructive

(49, 70, 49666)-Net over F32 — Digital

(49, 70, large)-Net in Base 32 — Upper bound on s

(49, 70, 3280)-Net over F₃₂ — Constructive and digital

(49, 70, 49666)-Net over F₃₂ — Digital