MinT - Best Known (78−35, 78, s)-Nets in Base 32

Best Known (78−35, 78, s)-Nets in Base 32

(78−35, 78, 240)-Net over F₃₂ — Constructive and digital

Digital (43, 78, 240)-net over F₃₂, using

7 times m-reduction [i] based on digital (43, 85, 240)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(78−35, 78, 513)-Net in Base 32 — Constructive

(43, 78, 513)-net in base 32, using

12 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
- base change [i] based on digital (28, 75, 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]

(78−35, 78, 1266)-Net over F₃₂ — Digital

Digital (43, 78, 1266)-net over F₃₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(32⁷⁸, 1266, F₃₂, 35) (dual of [1266, 1188, 36]-code), using
- 228 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 0, 1, 0, 0, 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 62 times 0, 1, 94 times 0) [i] based on linear OA(32⁶⁶, 1026, F₃₂, 35) (dual of [1026, 960, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(33) [i] based on
    1. linear OA(32⁶⁶, 1024, F₃₂, 35) (dual of [1024, 958, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(32⁶⁴, 1024, F₃₂, 34) (dual of [1024, 960, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 32²âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(32⁰, 2, F₃₂, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(78−35, 78, 1520635)-Net in Base 32 — Upper bound on s

There is no (43, 78, 1520636)-net in base 32, because

1 times m-reduction [i] would yield (43, 77, 1520636)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 78 804558 132894 957076 894228 351993 379002 401645 070530 082765 263328 882417 280644 683950 766818 417514 101407 191255 519382 123297 > 32⁷⁷ [i]