MinT - Best Known (122, 122+137, s)-Nets in Base 4

Best Known (122, 122+137, s)-Nets in Base 4

(122, 122+137, 130)-Net over F₄ — Constructive and digital

Digital (122, 259, 130)-net over F₄, using

t-expansion [i] based on digital (105, 259, 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]

(122, 122+137, 168)-Net over F₄ — Digital

Digital (122, 259, 168)-net over F₄, using

t-expansion [i] based on digital (115, 259, 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]

(122, 122+137, 1622)-Net in Base 4 — Upper bound on s

There is no (122, 259, 1623)-net in base 4, because

1 times m-reduction [i] would yield (122, 258, 1623)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 220679 005500 065312 084438 273013 087425 711421 029492 942403 373132 229278 675836 157601 543227 836844 863125 901920 568534 559647 950974 582390 924031 210906 167551 091802 434495 > 4²⁵⁸ [i]