MinT - Best Known (180−21, 180, s)-Nets in Base 4

Best Known (180−21, 180, s)-Nets in Base 4

(180−21, 180, 419445)-Net over F₄ — Constructive and digital

Digital (159, 180, 419445)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (145, 166, 419430)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁶⁶, 419430, F₄, 21, 21) (dual of [(419430, 21), 8807864, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(4¹⁶⁶, 4194301, F₄, 21) (dual of [4194301, 4194135, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶⁶, 4194304, F₄, 21) (dual of [4194304, 4194138, 22]-code), using
        an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(180−21, 180, 2097186)-Net over F₄ — Digital

Digital (159, 180, 2097186)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4¹⁸⁰, 2097186, F₄, 2, 21) (dual of [(2097186, 2), 4194192, 22]-NRT-code), using
- 4¹ times duplication [i] based on linear OOA(4¹⁷⁹, 2097186, F₄, 2, 21) (dual of [(2097186, 2), 4194193, 22]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(4¹⁷⁹, 4194372, F₄, 21) (dual of [4194372, 4194193, 22]-code), using
    - constructionÂ X applied to C_e(20) âŠ‚ C_e(13) [i] based on
      1. linear OA(4¹⁶⁶, 4194304, F₄, 21) (dual of [4194304, 4194138, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      2. linear OA(4¹¹¹, 4194304, F₄, 14) (dual of [4194304, 4194193, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 4194303Â = 4¹¹âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      3. linear OA(4¹³, 68, F₄, 6) (dual of [68, 55, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹³, 70, F₄, 6) (dual of [70, 57, 7]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(180−21, 180, large)-Net in Base 4 — Upper bound on s

There is no (159, 180, large)-net in base 4, because

19 times m-reduction [i] would yield (159, 161, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]