MinT - Best Known (242−123, 242, s)-Nets in Base 4

Best Known (242−123, 242, s)-Nets in Base 4

(242−123, 242, 130)-Net over F₄ — Constructive and digital

Digital (119, 242, 130)-net over F₄, using

t-expansion [i] based on digital (105, 242, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(242−123, 242, 168)-Net over F₄ — Digital

Digital (119, 242, 168)-net over F₄, using

t-expansion [i] based on digital (115, 242, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(242−123, 242, 1828)-Net in Base 4 — Upper bound on s

There is no (119, 242, 1829)-net in base 4, because

1 times m-reduction [i] would yield (119, 241, 1829)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 763279 293839 333227 773520 815311 591869 807333 097889 723953 430025 590793 444183 817932 369147 930537 289732 482489 560225 483460 525336 161283 006000 264164 519040 > 4²⁴¹ [i]