MinT - Best Known (123−26, 123, s)-Nets in Base 4

Best Known (123−26, 123, s)-Nets in Base 4

(123−26, 123, 1045)-Net over F₄ — Constructive and digital

Digital (97, 123, 1045)-net over F₄, using

4¹ times duplication [i] based on digital (96, 122, 1045)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 18, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (78, 104, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 26, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(123−26, 123, 3737)-Net over F₄ — Digital

Digital (97, 123, 3737)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²³, 3737, F₄, 26) (dual of [3737, 3614, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²³, 4128, F₄, 26) (dual of [4128, 4005, 27]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹²², 4127, F₄, 26) (dual of [4127, 4005, 27]-code), using
    - constructionÂ X applied to C_e(25) âŠ‚ C_e(20) [i] based on
      1. linear OA(4¹¹⁵, 4096, F₄, 26) (dual of [4096, 3981, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      2. linear OA(4⁹¹, 4096, F₄, 21) (dual of [4096, 4005, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(4⁷, 31, F₄, 4) (dual of [31, 24, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(123−26, 123, 939061)-Net in Base 4 — Upper bound on s

There is no (97, 123, 939062)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 113 079424 244212 126682 572726 043354 952494 996488 483883 900849 961896 930452 643914 > 4¹²³ [i]