MinT - Best Known (70, 242, s)-Nets in Base 4

Best Known (70, 242, s)-Nets in Base 4

(70, 242, 66)-Net over F₄ — Constructive and digital

Digital (70, 242, 66)-net over F₄, using

t-expansion [i] based on digital (49, 242, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(70, 242, 105)-Net over F₄ — Digital

Digital (70, 242, 105)-net over F₄, using

net from sequence [i] based on digital (70, 104)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
  - Drinfeld modules of rank 1 [i]

(70, 242, 434)-Net over F₄ — Upper bound on s (digital)

There is no digital (70, 242, 435)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴², 435, F₄, 172) (dual of [435, 193, 173]-code), but
- residual code [i] would yield OA(4⁷⁰, 262, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 45 983165 517068 297448 511524 093141 662826 825837 816964 700852 432473 444189 329902 728330 463517 081600 / 32 437855 549903 223644 246089 638836 795109 698224 516577 > 4⁷⁰ [i]

(70, 242, 473)-Net in Base 4 — Upper bound on s

There is no (70, 242, 474)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 57 584941 843380 266709 665657 913629 618844 694704 104587 021037 318933 043416 272140 769518 827921 504637 205715 488000 363282 988597 424276 011981 965902 784119 753796 > 4²⁴² [i]