MinT - Best Known (230−164, 230, s)-Nets in Base 4

Best Known (230−164, 230, s)-Nets in Base 4

(230−164, 230, 66)-Net over F₄ — Constructive and digital

Digital (66, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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]

(230−164, 230, 99)-Net over F₄ — Digital

Digital (66, 230, 99)-net over F₄, using

t-expansion [i] based on digital (61, 230, 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]

(230−164, 230, 403)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 230, 404)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁰, 404, F₄, 164) (dual of [404, 174, 165]-code), but
- residual code [i] would yield OA(4⁶⁶, 239, S₄, 41), but
  - the linear programming bound shows that MÂ â‰¥ 222 701100 724979 298348 829615 805692 012896 998205 225198 824371 851659 244379 751219 018386 636800 / 38358 477730 413015 488846 745362 992009 108847 310917 > 4⁶⁶ [i]

(230−164, 230, 445)-Net in Base 4 — Upper bound on s

There is no (66, 230, 446)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 241620 881798 888221 700598 728457 836295 103587 326822 119860 421455 208712 245928 190857 166618 729444 775541 446290 488732 855861 280040 565953 136406 049060 > 4²³⁰ [i]