MinT - Best Known (62, 62+160, s)-Nets in Base 4

Best Known (62, 62+160, s)-Nets in Base 4

(62, 62+160, 66)-Net over F₄ — Constructive and digital

Digital (62, 222, 66)-net over F₄, using

t-expansion [i] based on digital (49, 222, 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]

(62, 62+160, 99)-Net over F₄ — Digital

Digital (62, 222, 99)-net over F₄, using

t-expansion [i] based on digital (61, 222, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(62, 62+160, 351)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 222, 352)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²², 352, F₄, 160) (dual of [352, 130, 161]-code), but
- residual code [i] would yield OA(4⁶², 191, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 6 721352 642212 013290 164202 930663 188729 706683 395933 433874 909615 380474 388774 465686 228242 006016 / 300843 081103 152389 559805 379819 997539 174337 778277 134589 > 4⁶² [i]

(62, 62+160, 414)-Net in Base 4 — Upper bound on s

There is no (62, 222, 415)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 955264 568487 814613 118464 035157 931206 131110 456132 013771 085334 668230 751518 102925 324196 907498 071813 747809 634025 060553 151485 956148 942110 > 4²²² [i]