MinT - Best Known (61, 61+180, s)-Nets in Base 4

Best Known (61, 61+180, s)-Nets in Base 4

(61, 61+180, 66)-Net over F₄ — Constructive and digital

Digital (61, 241, 66)-net over F₄, using

t-expansion [i] based on digital (49, 241, 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]

(61, 61+180, 99)-Net over F₄ — Digital

Digital (61, 241, 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]

(61, 61+180, 257)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 241, 258)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴¹, 258, F₄, 180) (dual of [258, 17, 181]-code), but
- residual code [i] would yield OA(4⁶¹, 77, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 87771 116150 262149 392012 316054 470316 656445 882368 / 16226 709967 > 4⁶¹ [i]

(61, 61+180, 397)-Net in Base 4 — Upper bound on s

There is no (61, 241, 398)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 125067 212473 667492 686137 287474 258662 304585 949185 608394 221677 134010 345812 215881 917691 632683 635214 516260 282435 527422 777851 367058 709017 576218 068144 > 4²⁴¹ [i]