MinT - Best Known (65, 65+172, s)-Nets in Base 4

Best Known (65, 65+172, s)-Nets in Base 4

(65, 65+172, 66)-Net over F₄ — Constructive and digital

Digital (65, 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]

(65, 65+172, 99)-Net over F₄ — Digital

Digital (65, 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]

(65, 65+172, 346)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 237, 347)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁷, 347, F₄, 172) (dual of [347, 110, 173]-code), but
- residual code [i] would yield OA(4⁶⁵, 174, S₄, 43), but
  - the linear programming bound shows that MÂ â‰¥ 2 356982 708196 944001 632405 639197 943867 825680 395816 503599 985988 433957 852565 175436 909425 167874 773494 005760 / 1698 395268 555289 237515 503113 977957 762702 705839 816361 669722 314819 > 4⁶⁵ [i]

(65, 65+172, 431)-Net in Base 4 — Upper bound on s

There is no (65, 237, 432)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53344 604291 530357 755483 388900 991278 986261 512957 907624 449549 942728 769678 861570 392312 457374 291776 191842 811855 639580 838329 176944 615673 351186 126260 > 4²³⁷ [i]