MinT - Best Known (87−23, 87, s)-Nets in Base 4

Best Known (87−23, 87, s)-Nets in Base 4

(87−23, 87, 384)-Net over F₄ — Constructive and digital

Digital (64, 87, 384)-net over F₄, using

t-expansion [i] based on digital (63, 87, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 29, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(87−23, 87, 450)-Net in Base 4 — Constructive

(64, 87, 450)-net in base 4, using

4³ times duplication [i] based on (61, 84, 450)-net in base 4, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
  - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
    - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(87−23, 87, 830)-Net over F₄ — Digital

Digital (64, 87, 830)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁷, 830, F₄, 23) (dual of [830, 743, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸⁷, 1035, F₄, 23) (dual of [1035, 948, 24]-code), using
  - constructionÂ X applied to C_e(22) âŠ‚ C_e(20) [i] based on
    1. linear OA(4⁸⁶, 1024, F₄, 23) (dual of [1024, 938, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
    2. linear OA(4⁷⁶, 1024, F₄, 21) (dual of [1024, 948, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    3. linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(87−23, 87, 83341)-Net in Base 4 — Upper bound on s

There is no (64, 87, 83342)-net in base 4, because

1 times m-reduction [i] would yield (64, 86, 83342)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5986 567995 088761 460857 754774 652503 647718 117387 684210 > 4⁸⁶ [i]