MinT - Best Known (56, 56+53, s)-Nets in Base 32

Best Known (56, 56+53, s)-Nets in Base 32

(56, 56+53, 240)-Net over F₃₂ — Constructive and digital

Digital (56, 109, 240)-net over F₃₂, using

t-expansion [i] based on digital (51, 109, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 40, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]
  2. digital (11, 69, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(56, 56+53, 513)-Net in Base 32 — Constructive

(56, 109, 513)-net in base 32, using

32¹ times duplication [i] based on (55, 108, 513)-net in base 32, using
- t-expansion [i] based on (46, 108, 513)-net in base 32, using
  - base change [i] based on digital (28, 90, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(56, 56+53, 962)-Net over F₃₂ — Digital

Digital (56, 109, 962)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32¹⁰⁹, 962, F₃₂, 53) (dual of [962, 853, 54]-code), using
- discarding factors / shortening the dual code based on linear OA(32¹⁰⁹, 1047, F₃₂, 53) (dual of [1047, 938, 54]-code), using
  - constructionÂ X applied to C_e(52) âŠ‚ C_e(44) [i] based on
    1. linear OA(32¹⁰², 1024, F₃₂, 53) (dual of [1024, 922, 54]-code), using an extension C_e(52) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,52], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 53 [i]
    2. linear OA(32⁸⁶, 1024, F₃₂, 45) (dual of [1024, 938, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
    3. linear OA(32⁷, 23, F₃₂, 7) (dual of [23, 16, 8]-code or 23-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]

(56, 56+53, 608251)-Net in Base 32 — Upper bound on s

There is no (56, 109, 608252)-net in base 32, because

1 times m-reduction [i] would yield (56, 108, 608252)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 599190 353737 908233 828342 698202 770358 978829 185566 578186 566203 851041 645192 195175 828547 138142 319788 147566 738434 923311 906564 594469 595273 162633 277213 928993 981651 943181 > 32¹⁰⁸ [i]