MinT - Best Known (63, 219, s)-Nets in Base 4

Best Known (63, 219, s)-Nets in Base 4

(63, 219, 66)-Net over F₄ — Constructive and digital

Digital (63, 219, 66)-net over F₄, using

t-expansion [i] based on digital (49, 219, 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]

(63, 219, 99)-Net over F₄ — Digital

Digital (63, 219, 99)-net over F₄, using

t-expansion [i] based on digital (61, 219, 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]

(63, 219, 389)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 219, 390)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹⁹, 390, F₄, 156) (dual of [390, 171, 157]-code), but
- residual code [i] would yield OA(4⁶³, 233, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 2 394144 335707 471520 884224 011051 775831 492539 148504 246999 784192 681791 545828 966400 000000 / 26753 789254 137656 696530 296222 884590 520869 895903 > 4⁶³ [i]

(63, 219, 426)-Net in Base 4 — Upper bound on s

There is no (63, 219, 427)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 776263 120125 645721 248014 212929 566602 588070 065993 785620 591552 596172 094555 758572 217442 106850 909221 485474 973364 979535 797587 662650 466480 > 4²¹⁹ [i]