MinT - Best Known (243−172, 243, s)-Nets in Base 4

Best Known (243−172, 243, s)-Nets in Base 4

(243−172, 243, 66)-Net over F₄ — Constructive and digital

Digital (71, 243, 66)-net over F₄, using

t-expansion [i] based on digital (49, 243, 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]

(243−172, 243, 105)-Net over F₄ — Digital

Digital (71, 243, 105)-net over F₄, using

t-expansion [i] based on digital (70, 243, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(243−172, 243, 453)-Net over F₄ — Upper bound on s (digital)

There is no digital (71, 243, 454)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴³, 454, F₄, 172) (dual of [454, 211, 173]-code), but
- residual code [i] would yield OA(4⁷¹, 281, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 373 183646 729416 144099 480364 087718 649481 545879 174441 862191 129687 019685 715860 929497 289195 520000 / 66 442053 478467 341276 388448 356669 192467 605369 216717 > 4⁷¹ [i]

(243−172, 243, 481)-Net in Base 4 — Upper bound on s

There is no (71, 243, 482)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204 573275 007112 518974 334446 565340 106324 173618 566762 650384 476202 085315 401489 500891 194464 590563 337212 379565 850447 524365 964281 278771 787639 101743 118160 > 4²⁴³ [i]