MinT - Best Known (49−13, 49, s)-Nets in Base 4

Best Known (49−13, 49, s)-Nets in Base 4

(49−13, 49, 312)-Net over F₄ — Constructive and digital

Digital (36, 49, 312)-net over F₄, using

4¹ times duplication [i] based on digital (35, 48, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 16, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(49−13, 49, 387)-Net in Base 4 — Constructive

(36, 49, 387)-net in base 4, using

4¹ times duplication [i] based on (35, 48, 387)-net in base 4, using
- trace code for nets [i] based on (3, 16, 129)-net in base 64, using
  - 5 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
    - base change [i] based on digital (0, 18, 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]

(49−13, 49, 686)-Net over F₄ — Digital

Digital (36, 49, 686)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁹, 686, F₄, 13) (dual of [686, 637, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴⁹, 1037, F₄, 13) (dual of [1037, 988, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
    1. linear OA(4⁴⁶, 1024, F₄, 13) (dual of [1024, 978, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(4³⁶, 1024, F₄, 10) (dual of [1024, 988, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(4³, 13, F₄, 2) (dual of [13, 10, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(49−13, 49, 65395)-Net in Base 4 — Upper bound on s

There is no (36, 49, 65396)-net in base 4, because

1 times m-reduction [i] would yield (36, 48, 65396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 79228 500890 750128 800650 565604 > 4⁴⁸ [i]