MinT - Best Known (72, 252, s)-Nets in Base 4

Best Known (72, 252, s)-Nets in Base 4

(72, 252, 66)-Net over F₄ — Constructive and digital

Digital (72, 252, 66)-net over F₄, using

t-expansion [i] based on digital (49, 252, 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]

(72, 252, 105)-Net over F₄ — Digital

Digital (72, 252, 105)-net over F₄, using

t-expansion [i] based on digital (70, 252, 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]

(72, 252, 430)-Net over F₄ — Upper bound on s (digital)

There is no digital (72, 252, 431)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵², 431, F₄, 180) (dual of [431, 179, 181]-code), but
- residual code [i] would yield OA(4⁷², 250, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 844 253129 128337 968002 738138 104958 795392 851484 110411 486022 086287 533951 460726 969110 480212 292670 586880 / 36 177347 187970 342236 372937 633185 534214 836452 245739 305493 > 4⁷² [i]

(72, 252, 483)-Net in Base 4 — Upper bound on s

There is no (72, 252, 484)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 56 332233 132115 683222 417384 859681 112549 923979 206124 985275 527783 931325 902089 902978 230428 725018 884021 068138 503196 516085 513592 053699 815758 202309 802871 482068 > 4²⁵² [i]