MinT - Best Known (26, 26+10, s)-Nets in Base 4

Best Known (26, 26+10, s)-Nets in Base 4

(26, 26+10, 240)-Net over F₄ — Constructive and digital

Digital (26, 36, 240)-net over F₄, using

t-expansion [i] based on digital (25, 36, 240)-net over F₄, using
- trace code for nets [i] based on digital (1, 12, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(26, 26+10, 387)-Net in Base 4 — Constructive

(26, 36, 387)-net in base 4, using

trace code for nets [i] based on (2, 12, 129)-net in base 64, using
- 2 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
  - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(26, 26+10, 535)-Net over F₄ — Digital

Digital (26, 36, 535)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³⁶, 535, F₄, 10) (dual of [535, 499, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4³⁶, 1023, F₄, 10) (dual of [1023, 987, 11]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(26, 26+10, 18769)-Net in Base 4 — Upper bound on s

There is no (26, 36, 18770)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4722 484125 336445 479748 > 4³⁶ [i]