MinT - Best Known (61−12, 61, s)-Nets in Base 4

Best Known (61−12, 61, s)-Nets in Base 4

(61−12, 61, 1063)-Net over F₄ — Constructive and digital

Digital (49, 61, 1063)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 13, 35)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 17)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁴, 17, F₄, 3, 3) (dual of [(17, 3), 47, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(4⁴, 17, F₄, 2, 3) (dual of [(17, 2), 30, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4⁴, 17, F₄, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
        ovoid in PG(3,Â 4) [i]
    2. digital (3, 9, 18)-net over F₄, using
      - linear code embedding found in a computer search [i]
2. digital (36, 48, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 12, 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]

(61−12, 61, 4330)-Net over F₄ — Digital

Digital (49, 61, 4330)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁶¹, 4330, F₄, 12) (dual of [4330, 4269, 13]-code), using
- 228 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 5 times 0, 1, 14 times 0, 1, 29 times 0, 1, 59 times 0, 1, 113 times 0) [i] based on linear OA(4⁵⁴, 4095, F₄, 12) (dual of [4095, 4041, 13]-code), using
  - 1 times truncation [i] based on linear OA(4⁵⁵, 4096, F₄, 13) (dual of [4096, 4041, 14]-code), using
    - an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 4⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(61−12, 61, 1318386)-Net in Base 4 — Upper bound on s

There is no (49, 61, 1318387)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 316935 574884 074813 206716 306513 933980 > 4⁶¹ [i]