MinT - Best Known (59, 59+148, s)-Nets in Base 4

Best Known (59, 59+148, s)-Nets in Base 4

(59, 59+148, 66)-Net over F₄ — Constructive and digital

Digital (59, 207, 66)-net over F₄, using

t-expansion [i] based on digital (49, 207, 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]

(59, 59+148, 91)-Net over F₄ — Digital

Digital (59, 207, 91)-net over F₄, using

t-expansion [i] based on digital (50, 207, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(59, 59+148, 357)-Net over F₄ — Upper bound on s (digital)

There is no digital (59, 207, 358)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁷, 358, F₄, 148) (dual of [358, 151, 149]-code), but
- residual code [i] would yield OA(4⁵⁹, 209, S₄, 37), but
  - the linear programming bound shows that MÂ â‰¥ 3446 699662 509398 967731 365644 377802 984275 634503 925011 721737 994240 000000 000000 / 10213 023598 987442 955247 667333 699389 861739 > 4⁵⁹ [i]

(59, 59+148, 398)-Net in Base 4 — Upper bound on s

There is no (59, 207, 399)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43503 865281 383604 865464 923485 642113 488118 508063 709307 399425 153628 369012 475870 857438 936238 946987 619712 891247 470129 250311 024114 > 4²⁰⁷ [i]