MinT - Best Known (69, 69+168, s)-Nets in Base 4

Best Known (69, 69+168, s)-Nets in Base 4

(69, 69+168, 66)-Net over F₄ — Constructive and digital

Digital (69, 237, 66)-net over F₄, using

t-expansion [i] based on digital (49, 237, 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]

(69, 69+168, 99)-Net over F₄ — Digital

Digital (69, 237, 99)-net over F₄, using

t-expansion [i] based on digital (61, 237, 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]

(69, 69+168, 433)-Net over F₄ — Upper bound on s (digital)

There is no digital (69, 237, 434)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁷, 434, F₄, 168) (dual of [434, 197, 169]-code), but
- residual code [i] would yield OA(4⁶⁹, 265, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 52348 985175 743312 195164 699043 560255 242722 252419 549522 172173 323504 377479 561128 758476 800000 / 141682 475264 470819 104457 406563 443902 424070 593403 > 4⁶⁹ [i]

(69, 69+168, 467)-Net in Base 4 — Upper bound on s

There is no (69, 237, 468)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48944 151070 470910 340772 242273 222234 911108 002271 306343 572321 313506 951281 671072 147778 977075 837629 277964 480101 229635 534203 251774 629439 829540 094416 > 4²³⁷ [i]