MinT - Best Known (72, 72+21, s)-Nets in Base 32

Best Known (72, 72+21, s)-Nets in Base 32

(72, 72+21, 104902)-Net over F₃₂ — Constructive and digital

Digital (72, 93, 104902)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (61, 82, 104858)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸², 104858, F₃₂, 21, 21) (dual of [(104858, 21), 2201936, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(32⁸², 1048581, F₃₂, 21) (dual of [1048581, 1048499, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁸², 1048586, F₃₂, 21) (dual of [1048586, 1048504, 22]-code), using
        constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
        linear OA(32⁸¹, 1048577, F₃₂, 21) (dual of [1048577, 1048496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(32⁷³, 1048577, F₃₂, 19) (dual of [1048577, 1048504, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 32⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [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]

(72, 72+21, 209717)-Net in Base 32 — Constructive

(72, 93, 209717)-net in base 32, using

32² times duplication [i] based on (70, 91, 209717)-net in base 32, using
- base change [i] based on digital (44, 65, 209717)-net over F₁₂₈, using
  - net defined by OOA [i] based on linear OOA(128⁶⁵, 209717, F₁₂₈, 21, 21) (dual of [(209717, 21), 4403992, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(128⁶⁵, 2097171, F₁₂₈, 21) (dual of [2097171, 2097106, 22]-code), using
      - constructionÂ X applied to C_e(20) âŠ‚ C_e(15) [i] based on
        linear OA(128⁶¹, 2097152, F₁₂₈, 21) (dual of [2097152, 2097091, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
        linear OA(128⁴⁶, 2097152, F₁₂₈, 16) (dual of [2097152, 2097106, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2097151Â = 128³âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
        linear OA(128⁴, 19, F₁₂₈, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,128)), using
        discarding factors / shortening the dual code based on linear OA(128⁴, 128, F₁₂₈, 4) (dual of [128, 124, 5]-code or 128-arc in PG(3,128)), using
        Reedâ€“Solomon code RS(124,128) [i]

(72, 72+21, 2672351)-Net over F₃₂ — Digital

Digital (72, 93, 2672351)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+21, large)-Net in Base 32 — Upper bound on s

There is no (72, 93, large)-net in base 32, because

19 times m-reduction [i] would yield (72, 74, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (72, 72+21, s)-Nets in Base 32

(72, 72+21, 104902)-Net over F32 — Constructive and digital

(72, 72+21, 209717)-Net in Base 32 — Constructive

(72, 72+21, 2672351)-Net over F32 — Digital

(72, 72+21, large)-Net in Base 32 — Upper bound on s

(72, 72+21, 104902)-Net over F₃₂ — Constructive and digital

(72, 72+21, 2672351)-Net over F₃₂ — Digital