MinT - Best Known (213−154, 213, s)-Nets in Base 4

Best Known (213−154, 213, s)-Nets in Base 4

(213−154, 213, 66)-Net over F₄ — Constructive and digital

Digital (59, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 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]

(213−154, 213, 91)-Net over F₄ — Digital

Digital (59, 213, 91)-net over F₄, using

t-expansion [i] based on digital (50, 213, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(213−154, 213, 338)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 213, 339)-net over F₄, because

2 times m-reduction [i] would yield digital (59, 211, 339)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹¹, 339, F₄, 152) (dual of [339, 128, 153]-code), but
  - residual code [i] would yield OA(4⁵⁹, 186, S₄, 38), but
    - the linear programming bound shows that MÂ â‰¥ 458116 895032 657940 162336 762311 305276 187123 578098 709475 576222 779546 009600 / 1 358774 703882 919180 099545 580638 520853 > 4⁵⁹ [i]

(213−154, 213, 394)-Net in Base 4 — Upper bound on s

There is no (59, 213, 395)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 190 557211 434649 546376 689750 941389 214524 730539 047242 211983 202136 807836 862148 536710 930705 217766 129275 562484 379038 566088 724568 318784 > 4²¹³ [i]