MinT - Best Known (151−32, 151, s)-Nets in Base 4

Best Known (151−32, 151, s)-Nets in Base 4

(151−32, 151, 1049)-Net over F₄ — Constructive and digital

Digital (119, 151, 1049)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 23, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (96, 128, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 32, 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]

(151−32, 151, 4088)-Net over F₄ — Digital

Digital (119, 151, 4088)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁵¹, 4088, F₄, 32) (dual of [4088, 3937, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁵¹, 4118, F₄, 32) (dual of [4118, 3967, 33]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(4¹⁴⁵, 4097, F₄, 33) (dual of [4097, 3952, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
    2. linear OA(4¹²¹, 4097, F₄, 27) (dual of [4097, 3976, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
    3. linear OA(4⁶, 21, F₄, 4) (dual of [21, 15, 5]-code), using
      - quaternary constacyclic codes with minimum distance 5 [i]

(151−32, 151, 1089816)-Net in Base 4 — Upper bound on s

There is no (119, 151, 1089817)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 148160 165819 181021 353628 248841 447852 162008 532817 356869 780370 025661 011798 457040 914184 379199 > 4¹⁵¹ [i]