MinT - Best Known (131−27, 131, s)-Nets in Base 4

Best Known (131−27, 131, s)-Nets in Base 4

(131−27, 131, 1055)-Net over F₄ — Constructive and digital

Digital (104, 131, 1055)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 23, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (81, 108, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 27, 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]

(131−27, 131, 4246)-Net over F₄ — Digital

Digital (104, 131, 4246)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³¹, 4246, F₄, 27) (dual of [4246, 4115, 28]-code), using
- 134 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0) [i] based on linear OA(4¹²¹, 4102, F₄, 27) (dual of [4102, 3981, 28]-code), using
  - constructionÂ X applied to C_e(26) âŠ‚ C_e(25) [i] based on
    1. linear OA(4¹²¹, 4096, F₄, 27) (dual of [4096, 3975, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    2. 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]
    3. linear OA(4⁰, 6, F₄, 0) (dual of [6, 6, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(131−27, 131, 1980991)-Net in Base 4 — Upper bound on s

There is no (104, 131, 1980992)-net in base 4, because

1 times m-reduction [i] would yield (104, 130, 1980992)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 852681 764494 311850 702123 689906 362417 227365 646309 352013 963025 728204 818093 310105 > 4¹³⁰ [i]