MinT - Best Known (238−172, 238, s)-Nets in Base 4

Best Known (238−172, 238, s)-Nets in Base 4

(238−172, 238, 66)-Net over F₄ — Constructive and digital

Digital (66, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 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]

(238−172, 238, 99)-Net over F₄ — Digital

Digital (66, 238, 99)-net over F₄, using

t-expansion [i] based on digital (61, 238, 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]

(238−172, 238, 363)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 238, 364)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁸, 364, F₄, 172) (dual of [364, 126, 173]-code), but
- residual code [i] would yield OA(4⁶⁶, 191, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 28 766535 420054 304848 397361 421684 986456 320614 942528 963784 752261 977481 718335 232020 438414 046131 126272 000000 / 5209 403279 129645 739501 754543 221181 850628 166933 877437 442223 389477 > 4⁶⁶ [i]

(238−172, 238, 439)-Net in Base 4 — Upper bound on s

There is no (66, 238, 440)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 211053 427698 261919 712471 800805 622204 634855 226895 521570 712311 635397 926126 083538 647125 254645 770996 280312 051358 891891 836424 276982 734307 941250 779950 > 4²³⁸ [i]