MinT - Best Known (63, 63+168, s)-Nets in Base 4

Best Known (63, 63+168, s)-Nets in Base 4

(63, 63+168, 66)-Net over F₄ — Constructive and digital

Digital (63, 231, 66)-net over F₄, using

t-expansion [i] based on digital (49, 231, 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, 63+168, 99)-Net over F₄ — Digital

Digital (63, 231, 99)-net over F₄, using

t-expansion [i] based on digital (61, 231, 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, 63+168, 333)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 231, 334)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³¹, 334, F₄, 168) (dual of [334, 103, 169]-code), but
- residual code [i] would yield OA(4⁶³, 165, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 34 077190 942729 726882 024839 335902 035522 717143 016283 935337 510998 524998 580004 361097 586089 334532 997120 / 383476 318008 814835 283839 224897 225684 554347 388608 827379 925139 > 4⁶³ [i]

(63, 63+168, 417)-Net in Base 4 — Upper bound on s

There is no (63, 231, 418)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 064885 957233 032058 461261 808557 891727 968534 792346 961640 045083 826422 410891 790645 940187 566181 808021 052731 091556 177919 464161 267189 208138 998224 > 4²³¹ [i]