MinT - Best Known (28, 115, s)-Nets in Base 5

Best Known (28, 115, s)-Nets in Base 5

(28, 115, 51)-Net over F₅ — Constructive and digital

Digital (28, 115, 51)-net over F₅, using

t-expansion [i] based on digital (22, 115, 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]

(28, 115, 55)-Net over F₅ — Digital

Digital (28, 115, 55)-net over F₅, using

t-expansion [i] based on digital (23, 115, 55)-net over F₅, using
- net from sequence [i] based on digital (23, 54)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 23 and N(F) ≥ 55, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(28, 115, 262)-Net over F₅ — Upper bound on s (digital)

There is no digital (28, 115, 263)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹¹⁵, 263, F₅, 87) (dual of [263, 148, 88]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(5¹¹⁴, 148, S₅, 87), but
    - the linear programming bound shows that MÂ â‰¥ 63687 687420 768027 997189 489566 120495 525254 388256 273970 899556 895034 800130 250829 397482 448257 505893 707275 390625 / 1273 099402 224877 219656 507008 > 5¹¹⁴ [i]
  2. linear OA(5¹⁴⁸, 263, F₅, 115) (dual of [263, 115, 116]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5¹⁴⁸, 250, F₅, 115) (dual of [250, 102, 116]-code), but
      - residual code [i] would yield OA(5³³, 134, S₅, 23), but
        the linear programming bound shows that MÂ â‰¥ 7 905984 850652 471068 879589 438438 415527 343750 / 66 073848 265952 909413 > 5³³ [i]

(28, 115, 270)-Net in Base 5 — Upper bound on s

There is no (28, 115, 271)-net in base 5, because

1 times m-reduction [i] would yield (28, 114, 271)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 52 971242 010190 248266 912629 393301 055022 486655 663951 858099 566190 446600 684102 552725 > 5¹¹⁴ [i]