MinT - Best Known (62, 62+176, s)-Nets in Base 4

Best Known (62, 62+176, s)-Nets in Base 4

(62, 62+176, 66)-Net over F₄ — Constructive and digital

Digital (62, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 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+176, 99)-Net over F₄ — Digital

Digital (62, 238, 99)-net over F₄, using

t-expansion [i] based on digital (61, 238, 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+176, 283)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 238, 284)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁸, 284, F₄, 176) (dual of [284, 46, 177]-code), but
- residual code [i] would yield OA(4⁶², 107, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 17622 602942 558177 133437 108590 576398 639747 070941 352929 752748 829629 227505 655642 080394 898835 505152 / 826 744207 227851 738915 816443 333621 359995 803052 054368 328125 > 4⁶² [i]

(62, 62+176, 406)-Net in Base 4 — Upper bound on s

There is no (62, 238, 407)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 218397 164864 941259 543818 803616 244810 398237 281163 202215 554025 874901 009846 849806 455934 022270 801442 032051 508585 207257 049825 697167 769026 121662 354480 > 4²³⁸ [i]