MinT - Best Known (89−23, 89, s)-Nets in Base 4

Best Known (89−23, 89, s)-Nets in Base 4

(89−23, 89, 384)-Net over F₄ — Constructive and digital

Digital (66, 89, 384)-net over F₄, using

t-expansion [i] based on digital (65, 89, 384)-net over F₄, using
- 1 times m-reduction [i] based on digital (65, 90, 384)-net over F₄, using
  - trace code for nets [i] based on digital (5, 30, 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]

(89−23, 89, 450)-Net in Base 4 — Constructive

(66, 89, 450)-net in base 4, using

1 times m-reduction [i] based on (66, 90, 450)-net in base 4, using
- trace code for nets [i] based on (6, 30, 150)-net in base 64, using
  - 5 times m-reduction [i] based on (6, 35, 150)-net in base 64, using
    - base change [i] based on digital (1, 30, 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]

(89−23, 89, 949)-Net over F₄ — Digital

Digital (66, 89, 949)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸⁹, 949, F₄, 23) (dual of [949, 860, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸⁹, 1038, F₄, 23) (dual of [1038, 949, 24]-code), using
  - constructionÂ XX applied to C_e(22) âŠ‚ C_e(20) âŠ‚ C_e(18) [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(4⁷¹, 1024, F₄, 19) (dual of [1024, 953, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    4. linear OA(4¹, 12, F₄, 1) (dual of [12, 11, 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]
    5. linear OA(4¹, 2, F₄, 1) (dual of [2, 1, 2]-code), using
      - dual of repetition code with length 2 [i]

(89−23, 89, 107235)-Net in Base 4 — Upper bound on s

There is no (66, 89, 107236)-net in base 4, because

1 times m-reduction [i] would yield (66, 88, 107236)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 95787 935342 473857 932147 215476 772560 996059 739320 369932 > 4⁸⁸ [i]

Best Known (89−23, 89, s)-Nets in Base 4

(89−23, 89, 384)-Net over F4 — Constructive and digital

(89−23, 89, 450)-Net in Base 4 — Constructive

(89−23, 89, 949)-Net over F4 — Digital

(89−23, 89, 107235)-Net in Base 4 — Upper bound on s

(89−23, 89, 384)-Net over F₄ — Constructive and digital

(89−23, 89, 949)-Net over F₄ — Digital