MinT - Best Known (125−96, 125, s)-Nets in Base 5

Best Known (125−96, 125, s)-Nets in Base 5

(125−96, 125, 51)-Net over F₅ — Constructive and digital

Digital (29, 125, 51)-net over F₅, using

t-expansion [i] based on digital (22, 125, 51)-net over F₅, using
- net from sequence [i] based on digital (22, 50)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 22 and N(F) ≥ 51, using
    - an explicitly constructive algebraic function field [i]

(125−96, 125, 56)-Net over F₅ — Digital

Digital (29, 125, 56)-net over F₅, using

net from sequence [i] based on digital (29, 55)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 29 and N(F) ≥ 56, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(125−96, 125, 205)-Net over F₅ — Upper bound on s (digital)

There is no digital (29, 125, 206)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹²⁵, 206, F₅, 96) (dual of [206, 81, 97]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹²⁴, 145, S₅, 96), but
    - the linear programming bound shows that MÂ â‰¥ 2 028644 498784 410275 247873 069358 447259 491535 870530 433923 990677 562432 249914 081694 441847 503185 272216 796875 / 3772 467794 619342 > 5¹²⁴ [i]
  2. OA(5⁸¹, 206, S₅, 61), but
    - discarding factors would yield OA(5⁸¹, 147, S₅, 61), but
      - the linear programming bound shows that MÂ â‰¥ 1801 967420 622368 689962 930247 526850 152973 257033 063732 781999 225904 261867 797465 849728 978516 974105 935698 388835 132593 511556 837544 758050 093116 004164 767413 481473 930913 954138 291648 314065 852879 858365 565783 139920 029538 955560 468346 632496 672299 339479 650370 776653 289794 921875 / 4 186643 178459 883219 702862 698615 277144 100265 674265 643053 114986 779948 703766 289759 321388 479230 390841 917294 621365 011643 746087 757152 860527 760881 609721 102126 183400 893270 911698 096754 097070 738492 675774 446837 523823 > 5⁸¹ [i]

(125−96, 125, 275)-Net in Base 5 — Upper bound on s

There is no (29, 125, 276)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2506 615644 593415 207697 707350 991531 248974 934611 156690 374776 356951 246830 252533 249087 258625 > 5¹²⁵ [i]